Properties

Label 2-756-21.20-c1-0-7
Degree $2$
Conductor $756$
Sign $0.377 + 0.925i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·5-s + (−1 − 2.44i)7-s − 4.24i·11-s + 2.44i·13-s + 1.73·17-s − 2.44i·19-s − 8.48i·23-s − 2.00·25-s − 4.24i·29-s + 7.34i·31-s + (−1.73 − 4.24i)35-s + 37-s − 1.73·41-s + 7·43-s + 12.1·47-s + ⋯
L(s)  = 1  + 0.774·5-s + (−0.377 − 0.925i)7-s − 1.27i·11-s + 0.679i·13-s + 0.420·17-s − 0.561i·19-s − 1.76i·23-s − 0.400·25-s − 0.787i·29-s + 1.31i·31-s + (−0.292 − 0.717i)35-s + 0.164·37-s − 0.270·41-s + 1.06·43-s + 1.76·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.377 + 0.925i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.377 + 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30026 - 0.873614i\)
\(L(\frac12)\) \(\approx\) \(1.30026 - 0.873614i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (1 + 2.44i)T \)
good5 \( 1 - 1.73T + 5T^{2} \)
11 \( 1 + 4.24iT - 11T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 - 1.73T + 17T^{2} \)
19 \( 1 + 2.44iT - 19T^{2} \)
23 \( 1 + 8.48iT - 23T^{2} \)
29 \( 1 + 4.24iT - 29T^{2} \)
31 \( 1 - 7.34iT - 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + 1.73T + 41T^{2} \)
43 \( 1 - 7T + 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 8.66T + 59T^{2} \)
61 \( 1 + 2.44iT - 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 + 8.48iT - 71T^{2} \)
73 \( 1 + 9.79iT - 73T^{2} \)
79 \( 1 - 5T + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 - 2.44iT - 97T^{2} \)
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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

−10.34426525859597704463551094021, −9.307913880278323664256075392099, −8.627052354356705930261600780148, −7.52686071648128751915038652967, −6.53208048026689920608092470947, −5.94167479771536022367367964805, −4.70091072826839444226978786153, −3.64766738488151746817079886887, −2.45577270716896707289794138655, −0.825239729247605122427609875143, 1.71418341039061882413093855062, 2.74665393252395637760540054506, 4.05125065144832968593126507141, 5.56490975815474180404195894810, 5.70997031704695878768360658482, 7.07667180901742417189476096528, 7.85769629247980859625780889293, 9.029108492514994409750931806313, 9.721644776378650718797070189679, 10.17610380946572966534954011485

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