# lfunc_search downloaded from the LMFDB on 12 April 2026. # Search link: https://www.lmfdb.org/L/rational/2/9126/1.1 # Query "{'degree': 2, 'conductor': 9126, 'rational': True}" returned 38 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, $\epsilon$, $r$, First zero, Origin] # For more details, see the definitions at the bottom of the file. "2-9126-1.1-c1-0-100" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.986484148115084291806338116091 ["EllipticCurve/Q/9126/w", "ModularForm/GL2/Q/holomorphic/9126/2/a/w/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/w"] "2-9126-1.1-c1-0-102" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.994383401249765949143925502647 ["EllipticCurve/Q/9126/bb", "ModularForm/GL2/Q/holomorphic/9126/2/a/bb/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/bb"] "2-9126-1.1-c1-0-109" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.02656732134574195310140915123 ["EllipticCurve/Q/9126/i", "ModularForm/GL2/Q/holomorphic/9126/2/a/i/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/i"] "2-9126-1.1-c1-0-11" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.36900810771020488643137882302 ["EllipticCurve/Q/9126/v", "ModularForm/GL2/Q/holomorphic/9126/2/a/v/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/v"] "2-9126-1.1-c1-0-117" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 1.06570387643644874631526716438 ["EllipticCurve/Q/9126/bk", "ModularForm/GL2/Q/holomorphic/9126/2/a/bk/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/bk"] "2-9126-1.1-c1-0-121" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 1.07970592529623201499951753996 ["EllipticCurve/Q/9126/o", "ModularForm/GL2/Q/holomorphic/9126/2/a/o/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/o"] "2-9126-1.1-c1-0-123" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.10055772945985914198664079254 ["EllipticCurve/Q/9126/t", "ModularForm/GL2/Q/holomorphic/9126/2/a/t/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/t"] "2-9126-1.1-c1-0-126" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.11785487112152252308308675853 ["EllipticCurve/Q/9126/k", "ModularForm/GL2/Q/holomorphic/9126/2/a/k/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/k"] "2-9126-1.1-c1-0-128" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 1.13041360550907771052716165364 ["EllipticCurve/Q/9126/bl", "ModularForm/GL2/Q/holomorphic/9126/2/a/bl/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/bl"] "2-9126-1.1-c1-0-149" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.24728285166735903607105073330 ["EllipticCurve/Q/9126/g", "ModularForm/GL2/Q/holomorphic/9126/2/a/g/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/g"] "2-9126-1.1-c1-0-158" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.31567889376404306432402580561 ["EllipticCurve/Q/9126/f", "ModularForm/GL2/Q/holomorphic/9126/2/a/f/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/f"] "2-9126-1.1-c1-0-159" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.31686288488810061713523678982 ["EllipticCurve/Q/9126/y", "ModularForm/GL2/Q/holomorphic/9126/2/a/y/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/y"] "2-9126-1.1-c1-0-161" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.33173223924086766131485229394 ["EllipticCurve/Q/9126/u", "ModularForm/GL2/Q/holomorphic/9126/2/a/u/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/u"] "2-9126-1.1-c1-0-17" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.47351710897760617833275411047 ["EllipticCurve/Q/9126/d", "ModularForm/GL2/Q/holomorphic/9126/2/a/d/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/d"] "2-9126-1.1-c1-0-172" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.40317373749104312020413431195 ["EllipticCurve/Q/9126/r", "ModularForm/GL2/Q/holomorphic/9126/2/a/r/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/r"] "2-9126-1.1-c1-0-178" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.43607901738286267894920040515 ["EllipticCurve/Q/9126/z", "ModularForm/GL2/Q/holomorphic/9126/2/a/z/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/z"] "2-9126-1.1-c1-0-179" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.44123634058884824335249942972 ["EllipticCurve/Q/9126/bg", "ModularForm/GL2/Q/holomorphic/9126/2/a/bg/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/bg"] "2-9126-1.1-c1-0-182" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.46301321623709505492726513841 ["EllipticCurve/Q/9126/q", "ModularForm/GL2/Q/holomorphic/9126/2/a/q/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/q"] "2-9126-1.1-c1-0-192" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.58375647677780296113843302736 ["EllipticCurve/Q/9126/be", "ModularForm/GL2/Q/holomorphic/9126/2/a/be/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/be"] "2-9126-1.1-c1-0-196" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.60193448766606386446184281695 ["EllipticCurve/Q/9126/bc", "ModularForm/GL2/Q/holomorphic/9126/2/a/bc/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/bc"] "2-9126-1.1-c1-0-201" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.65842835177173933370615256347 ["EllipticCurve/Q/9126/ba", "ModularForm/GL2/Q/holomorphic/9126/2/a/ba/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/ba"] "2-9126-1.1-c1-0-203" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.69998515585316327554617968183 ["EllipticCurve/Q/9126/bj", "ModularForm/GL2/Q/holomorphic/9126/2/a/bj/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/bj"] "2-9126-1.1-c1-0-36" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.57035172107307914961885360538 ["EllipticCurve/Q/9126/h", "ModularForm/GL2/Q/holomorphic/9126/2/a/h/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/h"] "2-9126-1.1-c1-0-38" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.57937171732173149179549233492 ["EllipticCurve/Q/9126/m", "ModularForm/GL2/Q/holomorphic/9126/2/a/m/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/m"] "2-9126-1.1-c1-0-41" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.61805458441454465019262436372 ["EllipticCurve/Q/9126/p", "ModularForm/GL2/Q/holomorphic/9126/2/a/p/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/p"] "2-9126-1.1-c1-0-44" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.63458757681007672229671322227 ["EllipticCurve/Q/9126/l", "ModularForm/GL2/Q/holomorphic/9126/2/a/l/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/l"] "2-9126-1.1-c1-0-50" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.68210997716964519883018895613 ["EllipticCurve/Q/9126/bf", "ModularForm/GL2/Q/holomorphic/9126/2/a/bf/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/bf"] "2-9126-1.1-c1-0-55" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.72297107319190299659796342805 ["EllipticCurve/Q/9126/n", "ModularForm/GL2/Q/holomorphic/9126/2/a/n/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/n"] "2-9126-1.1-c1-0-66" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.802956151487712417220650033167 ["EllipticCurve/Q/9126/bh", "ModularForm/GL2/Q/holomorphic/9126/2/a/bh/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/bh"] "2-9126-1.1-c1-0-7" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.26492081730174966263873226489 ["EllipticCurve/Q/9126/a", "ModularForm/GL2/Q/holomorphic/9126/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/a"] "2-9126-1.1-c1-0-70" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.823642156133624611839563201575 ["EllipticCurve/Q/9126/x", "ModularForm/GL2/Q/holomorphic/9126/2/a/x/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/x"] "2-9126-1.1-c1-0-8" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.30632681550368307894672755920 ["EllipticCurve/Q/9126/j", "ModularForm/GL2/Q/holomorphic/9126/2/a/j/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/j"] "2-9126-1.1-c1-0-81" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.871515025639793957514033854437 ["EllipticCurve/Q/9126/bi", "ModularForm/GL2/Q/holomorphic/9126/2/a/bi/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/bi"] "2-9126-1.1-c1-0-86" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.905776346274597090932365854893 ["EllipticCurve/Q/9126/b", "ModularForm/GL2/Q/holomorphic/9126/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/b"] "2-9126-1.1-c1-0-89" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.919633151002002238213231666565 ["EllipticCurve/Q/9126/c", "ModularForm/GL2/Q/holomorphic/9126/2/a/c/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/c"] "2-9126-1.1-c1-0-94" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.948304042763610562594737307119 ["EllipticCurve/Q/9126/bd", "ModularForm/GL2/Q/holomorphic/9126/2/a/bd/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/bd"] "2-9126-1.1-c1-0-98" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.976097226796189590991305885097 ["EllipticCurve/Q/9126/s", "ModularForm/GL2/Q/holomorphic/9126/2/a/s/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/s"] "2-9126-1.1-c1-0-99" 8.536479185508538 72.87147688462052 2 9126 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.985476849308814160206815194134 ["EllipticCurve/Q/9126/e", "ModularForm/GL2/Q/holomorphic/9126/2/a/e/1/1", "ModularForm/GL2/Q/holomorphic/9126/2/a/e"] # 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 $1$. #$\epsilon$ (root_number) -- # The **sign** of the functional equation of an analytic L-function, also called the **root number**, is the complex number $\varepsilon$ that appears in the functional equation of $\Lambda(s)=\varepsilon \overline{\Lambda}(1-s)$. The sign appears as the 4th entry in the quadruple # known as the Selberg data. #$r$ (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).