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Properties

Label 2-2541-231.101-c0-0-1
Degree $2$
Conductor $2541$
Sign $0.598 - 0.800i$
Analytic cond. $1.26812$
Root an. cond. $1.12611$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.994 − 0.104i)3-s + (−0.913 + 0.406i)4-s + (−0.933 − 0.358i)7-s + (0.978 − 0.207i)9-s + (−0.866 + 0.499i)12-s + (−0.596 + 1.83i)13-s + (0.669 − 0.743i)16-s + (1.29 + 0.575i)19-s + (−0.965 − 0.258i)21-s + (0.104 + 0.994i)25-s + (0.951 − 0.309i)27-s + (0.998 − 0.0523i)28-s + (0.743 − 0.669i)31-s + (−0.809 + 0.587i)36-s + (−0.181 + 1.72i)37-s + ⋯
L(s)  = 1  + (0.994 − 0.104i)3-s + (−0.913 + 0.406i)4-s + (−0.933 − 0.358i)7-s + (0.978 − 0.207i)9-s + (−0.866 + 0.499i)12-s + (−0.596 + 1.83i)13-s + (0.669 − 0.743i)16-s + (1.29 + 0.575i)19-s + (−0.965 − 0.258i)21-s + (0.104 + 0.994i)25-s + (0.951 − 0.309i)27-s + (0.998 − 0.0523i)28-s + (0.743 − 0.669i)31-s + (−0.809 + 0.587i)36-s + (−0.181 + 1.72i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $0.598 - 0.800i$
Analytic conductor: \(1.26812\)
Root analytic conductor: \(1.12611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2541} (2411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2541,\ (\ :0),\ 0.598 - 0.800i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.218607978\)
\(L(\frac12)\) \(\approx\) \(1.218607978\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.994 + 0.104i)T \)
7 \( 1 + (0.933 + 0.358i)T \)
11 \( 1 \)
good2 \( 1 + (0.913 - 0.406i)T^{2} \)
5 \( 1 + (-0.104 - 0.994i)T^{2} \)
13 \( 1 + (0.596 - 1.83i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.913 - 0.406i)T^{2} \)
19 \( 1 + (-1.29 - 0.575i)T + (0.669 + 0.743i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.743 + 0.669i)T + (0.104 - 0.994i)T^{2} \)
37 \( 1 + (0.181 - 1.72i)T + (-0.978 - 0.207i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - 0.517iT - T^{2} \)
47 \( 1 + (0.669 + 0.743i)T^{2} \)
53 \( 1 + (0.104 - 0.994i)T^{2} \)
59 \( 1 + (0.669 - 0.743i)T^{2} \)
61 \( 1 + (-0.346 + 0.384i)T + (-0.104 - 0.994i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.472 + 0.210i)T + (0.669 - 0.743i)T^{2} \)
79 \( 1 + (0.294 + 1.38i)T + (-0.913 + 0.406i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.64 + 0.535i)T + (0.809 + 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.377458074936541371044917841896, −8.530187080640321499971754100453, −7.69279453399527879965841331405, −7.11676010189590979927988674914, −6.33480663457276490847500085404, −5.00651982927059388555421573709, −4.24509928473034070855911034958, −3.53594544318395558290154662761, −2.79912213617015541831760998845, −1.41459151323040323462542605719, 0.807722655911062036215423290357, 2.52899208651637171009172153482, 3.18977072442004163778536564389, 4.04090851810480287418962049389, 5.12094195278489545002622054062, 5.63092115241284208526566495025, 6.78834903919022041116125991706, 7.64474322669028624443623147279, 8.348913476581426498015014781493, 9.013210470551115241414134305276

Graph of the $Z$-function along the critical line