# lfunc_search downloaded from the LMFDB on 21 September 2026. # Search link: https://www.lmfdb.org/L/1/161/161.146/r1-0 # Query "{'degree': 1, 'conductor': 161, 'spectral_label': 'r1-0'}" returned 52 lfunc_searchs, sorted by root analytic conductor. # Each entry in the following data list has the form: # [Label, $\alpha$, $A$, $d$, $N$, $\chi$, $\mu$, $\nu$, $w$, prim, arith, $\mathbb{Q}$, self-dual, $\operatorname{Arg}(\epsilon)$, $r$, First zero, Origin] # For more details, see the definitions at the bottom of the file. "1-161-161.101-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.101" [[1, 0.0]] [] 0 true true false false 0.13013749151086976 0 0.51729939015 ["Character/Dirichlet/161/101"] "1-161-161.102-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.102" [[1, 0.0]] [] 0 true true false false 0.38783178031491944 0 0.213758078029 ["Character/Dirichlet/161/102"] "1-161-161.104-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.104" [[1, 0.0]] [] 0 true true false false 0.03901128169859303 0 1.56302179084 ["Character/Dirichlet/161/104"] "1-161-161.107-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.107" [[1, 0.0]] [] 0 true true false false 0.02305667581426909 0 0.583631613106 ["Character/Dirichlet/161/107"] "1-161-161.108-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.108" [[1, 0.0]] [] 0 true true false false 0.17738958400369462 0 2.04358437821 ["Character/Dirichlet/161/108"] "1-161-161.109-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.109" [[1, 0.0]] [] 0 true true false false -0.4818974920947667 0 0.565053732595 ["Character/Dirichlet/161/109"] "1-161-161.11-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.11" [[1, 0.0]] [] 0 true true false false -0.3102609493650968 0 2.64342636419 ["Character/Dirichlet/161/11"] "1-161-161.110-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.110" [[1, 0.0]] [] 0 true true false false -0.13013749151086976 0 0.940714509547 ["Character/Dirichlet/161/110"] "1-161-161.114-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.114" [[1, 0.0]] [] 0 true true false false 0.0934198098603064 0 1.08806600304 ["Character/Dirichlet/161/114"] "1-161-161.117-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.117" [[1, 0.0]] [] 0 true true false false 0.44081576524746086 0 2.20323261776 ["Character/Dirichlet/161/117"] "1-161-161.118-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.118" [[1, 0.0]] [] 0 true true false false -0.3694895046800228 0 2.2923800393 ["Character/Dirichlet/161/118"] "1-161-161.12-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.12" [[1, 0.0]] [] 0 true true false false -0.4811616713553531 0 1.63232500492 ["Character/Dirichlet/161/12"] "1-161-161.124-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.124" [[1, 0.0]] [] 0 true true false false 0.08979158540297752 0 1.12037558266 ["Character/Dirichlet/161/124"] "1-161-161.13-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.13" [[1, 0.0]] [] 0 true true false false 0.34278336905025153 0 1.29035166786 ["Character/Dirichlet/161/13"] "1-161-161.130-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.130" [[1, 0.0]] [] 0 true true false false -0.1637829439063437 0 1.36114731044 ["Character/Dirichlet/161/130"] "1-161-161.131-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.131" [[1, 0.0]] [] 0 true true false false 0.1370436778958024 0 0.665410192635 ["Character/Dirichlet/161/131"] "1-161-161.135-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.135" [[1, 0.0]] [] 0 true true false false 0.1637829439063437 0 1.73154208191 ["Character/Dirichlet/161/135"] "1-161-161.137-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.137" [[1, 0.0]] [] 0 true true false false -0.0934198098603064 0 0.801040774143 ["Character/Dirichlet/161/137"] "1-161-161.146-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.146" [[1, 0.0]] [] 0 true true false false 0.3694895046800228 0 0.156153337657 ["Character/Dirichlet/161/146"] "1-161-161.149-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.149" [[1, 0.0]] [] 0 true true false false -0.12342132964448399 0 1.02272486505 ["Character/Dirichlet/161/149"] "1-161-161.150-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.150" [[1, 0.0]] [] 0 true true false false -0.44081576524746086 0 0.0918600646136 ["Character/Dirichlet/161/150"] "1-161-161.158-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.158" [[1, 0.0]] [] 0 true true false false -0.02305667581426909 0 0.85178989235 ["Character/Dirichlet/161/158"] "1-161-161.26-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.26" [[1, 0.0]] [] 0 true true false false 0.11033754226603112 0 1.00514216245 ["Character/Dirichlet/161/26"] "1-161-161.27-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.27" [[1, 0.0]] [] 0 true true false false -0.45853377855743904 0 1.17281996945 ["Character/Dirichlet/161/27"] "1-161-161.3-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.3" [[1, 0.0]] [] 0 true true false false 0.15068344837392336 0 0.0565840616725 ["Character/Dirichlet/161/3"] "1-161-161.30-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.30" [[1, 0.0]] [] 0 true true false false -0.38783178031491944 0 1.85055787992 ["Character/Dirichlet/161/30"] "1-161-161.31-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.31" [[1, 0.0]] [] 0 true true false false -0.11033754226603112 0 1.18967165611 ["Character/Dirichlet/161/31"] "1-161-161.37-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.37" [[1, 0.0]] [] 0 true true false false -0.2957152650798424 0 0.582226350707 ["Character/Dirichlet/161/37"] "1-161-161.41-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.41" [[1, 0.0]] [] 0 true true false false 0.39003546154307633 0 2.22078489489 ["Character/Dirichlet/161/41"] "1-161-161.44-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.44" [[1, 0.0]] [] 0 true true false false 0.3102609493650968 0 0.167692476172 ["Character/Dirichlet/161/44"] "1-161-161.48-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.48" [[1, 0.0]] [] 0 true true false false -0.03901128169859303 0 0.767540927512 ["Character/Dirichlet/161/48"] "1-161-161.51-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.51" [[1, 0.0]] [] 0 true true false false 0.4825548848004552 0 0.425444907387 ["Character/Dirichlet/161/51"] "1-161-161.52-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.52" [[1, 0.0]] [] 0 true true false false -0.021293268388614863 0 1.1424883941 ["Character/Dirichlet/161/52"] "1-161-161.53-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.53" [[1, 0.0]] [] 0 true true false false 0.4253285999644678 0 2.03928855678 ["Character/Dirichlet/161/53"] "1-161-161.54-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.54" [[1, 0.0]] [] 0 true true false false -0.15068344837392336 0 3.11313189825 ["Character/Dirichlet/161/54"] "1-161-161.55-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.55" [[1, 0.0]] [] 0 true true false false -0.39003546154307633 0 1.07273468229 ["Character/Dirichlet/161/55"] "1-161-161.59-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.59" [[1, 0.0]] [] 0 true true false false -0.1370436778958024 0 0.262555511322 ["Character/Dirichlet/161/59"] "1-161-161.6-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.6" [[1, 0.0]] [] 0 true true false false 0.45853377855743904 0 0.0153469610684 ["Character/Dirichlet/161/6"] "1-161-161.60-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.60" [[1, 0.0]] [] 0 true true false false -0.4825548848004552 0 2.06446199098 ["Character/Dirichlet/161/60"] "1-161-161.62-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.62" [[1, 0.0]] [] 0 true true false false -0.34278336905025153 0 0.458415464228 ["Character/Dirichlet/161/62"] "1-161-161.65-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.65" [[1, 0.0]] [] 0 true true false false 0.4818974920947667 0 2.66027749186 ["Character/Dirichlet/161/65"] "1-161-161.67-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.67" [[1, 0.0]] [] 0 true true false false 0.12342132964448399 0 1.36656509777 ["Character/Dirichlet/161/67"] "1-161-161.73-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.73" [[1, 0.0]] [] 0 true true false false 0.061639174496507104 0 1.68064778285 ["Character/Dirichlet/161/73"] "1-161-161.74-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.74" [[1, 0.0]] [] 0 true true false false 0.2957152650798424 0 0.908344417339 ["Character/Dirichlet/161/74"] "1-161-161.75-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.75" [[1, 0.0]] [] 0 true true false false -0.061639174496507104 0 0.872450097333 ["Character/Dirichlet/161/75"] "1-161-161.79-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.79" [[1, 0.0]] [] 0 true true false false -0.4253285999644678 0 0.587908013342 ["Character/Dirichlet/161/79"] "1-161-161.82-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.82" [[1, 0.0]] [] 0 true true false false -0.17738958400369462 0 1.41929851642 ["Character/Dirichlet/161/82"] "1-161-161.86-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.86" [[1, 0.0]] [] 0 true true false false -0.2950578723741539 0 1.79564896036 ["Character/Dirichlet/161/86"] "1-161-161.87-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.87" [[1, 0.0]] [] 0 true true false false -0.08979158540297752 0 0.727161141466 ["Character/Dirichlet/161/87"] "1-161-161.88-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.88" [[1, 0.0]] [] 0 true true false false 0.2950578723741539 0 0.046631738104 ["Character/Dirichlet/161/88"] "1-161-161.94-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.94" [[1, 0.0]] [] 0 true true false false 0.4811616713553531 0 0.244720143617 ["Character/Dirichlet/161/94"] "1-161-161.96-r1-0-0" 17.30185289193463 17.30185289193463 1 161 "161.96" [[1, 0.0]] [] 0 true true false false 0.021293268388614863 0 1.38030612345 ["Character/Dirichlet/161/96"] # Label -- # Each L-function $L$ has a label of the form d-N-q.k-x-y-i, where # * $d$ is the degree of $L$. # * $N$ is the conductor of $L$. When $N$ is a perfect power $m^n$ we write $N$ as $m$e$n$, since $N$ can be very large for some imprimitive L-functions. # * q.k is the label of the primitive Dirichlet character from which the central character is induced. # * x-y is the spectral label encoding the $\mu_j$ and $\nu_j$ in the analytically normalized functional equation. # * i is a non-negative integer disambiguating between L-functions that would otherwise have the same label. #$\alpha$ (root_analytic_conductor) -- # If $d$ is the degree of the L-function $L(s)$, the **root analytic conductor** $\alpha$ of $L$ is the $d$th root of the analytic conductor of $L$. It plays a role analogous to the root discriminant for number fields. #$A$ (analytic_conductor) -- # The **analytic conductor** of an L-function $L(s)$ with infinity factor $L_{\infty}(s)$ and conductor $N$ is the real number # \[ # A := \mathrm{exp}\left(2\mathrm{Re}\left(\frac{L_{\infty}'(1/2)}{L_{\infty}(1/2)}\right)\right)N. # \] #$d$ (degree) -- # The **degree** of an L-function is the number $J + 2K$ of Gamma factors occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The degree appears as the first component of the Selberg data of $L(s).$ In all known cases it is the degree of the polynomial of the inverse of the Euler factor at any prime not dividing the conductor. #$N$ (conductor) -- # The **conductor** of an L-function is the integer $N$ occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The conductor of an analytic L-function is the second component in the Selberg data. For a Dirichlet L-function # associated with a primitive Dirichlet character, the conductor of the L-function is the same as the conductor of the character. For a primitive L-function associated with a cusp form $\phi$ on $GL(2)/\mathbb Q$, the conductor of the L-function is the same as the level of $\phi$. # In the literature, the word _level_ is sometimes used instead of _conductor_. #$\chi$ (central_character) -- # An L-function has an Euler product of the form # $L(s) = \prod_p L_p(p^{-s})^{-1}$ # where $L_p(x) = 1 + a_p x + \ldots + (-1)^d \chi(p) x^d$. The character $\chi$ is a Dirichlet character mod $N$ and is called **central character** of the L-function. # Here, $N$ is the conductor of $L$. #$\mu$ (mus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$\nu$ (nus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$w$ (motivic_weight) -- # The **motivic weight** (or **arithmetic weight**) of an arithmetic L-function with analytic normalization $L_{an}(s)=\sum_{n=1}^\infty a_nn^{-s}$ is the least nonnegative integer $w$ for which $a_nn^{w/2}$ is an algebraic integer for all $n\ge 1$. # If the L-function arises from a motive, then the weight of the motive has the # same parity as the motivic weight of the L-function, but the weight of the motive # could be larger. This apparent discrepancy comes from the fact that a Tate twist # increases the weight of the motive. This corresponds to the change of variables # $s \mapsto s + j$ in the L-function of the motive. #prim (primitive) -- # An L-function is primitive if it cannot be written as a product of nontrivial L-functions. The "trivial L-function" is the constant function $1$. #arith (algebraic) -- # An L-function $L(s) = \sum_{n=1}^{\infty} a_n n^{-s}$ is called **arithmetic** if its Dirichlet coefficients $a_n$ are algebraic numbers. #$\mathbb{Q}$ (rational) -- # A **rational** L-function $L(s)$ is an arithmetic L-function with coefficient field $\Q$; equivalently, its Euler product in the arithmetic normalization can be written as a product over rational primes # \[ # L(s)=\prod_pL_p(p^{-s})^{-1} # \] # with $L_p\in \Z[T]$. #self-dual (self_dual) -- # An L-function $L(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ is called **self-dual** if its Dirichlet coefficients $a_n$ are real. #$\operatorname{Arg}(\epsilon)$ (root_angle) -- # The **root angle** of an L-function is the argument of its root number, as a real number $\alpha$ with $-0.5 < \alpha \le 0.5$. #$r$ (order_of_vanishing) -- # The **analytic rank** of an L-function $L(s)$ is its order of vanishing at its central point. # When the analytic rank $r$ is positive, the value listed in the LMFDB is typically an upper bound that is believed to be tight (in the sense that there are known to be $r$ zeroes located very near to the central point). #First zero (z1) -- # The **zeros** of an L-function $L(s)$ are the complex numbers $\rho$ for which $L(\rho)=0$. # Under the Riemann Hypothesis, every non-trivial zero $\rho$ lies on the critical line $\Re(s)=1/2$ (in the analytic normalization). # The **lowest zero** of an L-function $L(s)$ is the least $\gamma>0$ for which $L(1/2+i\gamma)=0$. Note that even when $L(1/2)=0$, the lowest zero is by definition a positive real number. #Origin (instance_urls) -- # L-functions arise from many different sources. Already in degree 2 we have examples of # L-functions associated with holomorphic cusp forms, with Maass forms, with elliptic curves, with characters of number fields (Hecke characters), and with 2-dimensional representations of the Galois group of a number field (Artin L-functions). # Sometimes an L-function may arise from more than one source. For example, the L-functions associated with elliptic curves are also associated with weight 2 cusp forms. A goal of the Langlands program ostensibly is to prove that any degree $d$ L-function is associated with an automorphic form on $\mathrm{GL}(d)$. Because of this representation theoretic genesis, one can associate an L-function not only to an automorphic representation but also to symmetric powers, or exterior powers of that representation, or to the tensor product of two representations (the Rankin-Selberg product of two L-functions).