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Properties

Label 2-1176-1176.317-c0-0-1
Degree $2$
Conductor $1176$
Sign $-0.934 - 0.355i$
Analytic cond. $0.586900$
Root an. cond. $0.766094$
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  + (0.0747 − 0.997i)2-s + (0.955 − 0.294i)3-s + (−0.988 − 0.149i)4-s + (−1.21 − 1.12i)5-s + (−0.222 − 0.974i)6-s + (−0.733 + 0.680i)7-s + (−0.222 + 0.974i)8-s + (0.826 − 0.563i)9-s + (−1.21 + 1.12i)10-s + (−1.63 − 1.11i)11-s + (−0.988 + 0.149i)12-s + (0.623 + 0.781i)14-s + (−1.48 − 0.716i)15-s + (0.955 + 0.294i)16-s + (−0.499 − 0.866i)18-s + ⋯
L(s)  = 1  + (0.0747 − 0.997i)2-s + (0.955 − 0.294i)3-s + (−0.988 − 0.149i)4-s + (−1.21 − 1.12i)5-s + (−0.222 − 0.974i)6-s + (−0.733 + 0.680i)7-s + (−0.222 + 0.974i)8-s + (0.826 − 0.563i)9-s + (−1.21 + 1.12i)10-s + (−1.63 − 1.11i)11-s + (−0.988 + 0.149i)12-s + (0.623 + 0.781i)14-s + (−1.48 − 0.716i)15-s + (0.955 + 0.294i)16-s + (−0.499 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.934 - 0.355i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ -0.934 - 0.355i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6872776321\)
\(L(\frac12)\) \(\approx\) \(0.6872776321\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0747 + 0.997i)T \)
3 \( 1 + (-0.955 + 0.294i)T \)
7 \( 1 + (0.733 - 0.680i)T \)
good5 \( 1 + (1.21 + 1.12i)T + (0.0747 + 0.997i)T^{2} \)
11 \( 1 + (1.63 + 1.11i)T + (0.365 + 0.930i)T^{2} \)
13 \( 1 + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + (0.733 + 0.680i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.733 - 0.680i)T^{2} \)
29 \( 1 + (-0.0931 - 0.116i)T + (-0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.955 + 1.65i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.955 + 0.294i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (0.988 + 0.149i)T^{2} \)
53 \( 1 + (-0.988 - 0.149i)T + (0.955 + 0.294i)T^{2} \)
59 \( 1 + (-0.733 + 0.680i)T + (0.0747 - 0.997i)T^{2} \)
61 \( 1 + (-0.955 + 0.294i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (0.0332 + 0.443i)T + (-0.988 + 0.149i)T^{2} \)
79 \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.134 + 0.0648i)T + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.365 + 0.930i)T^{2} \)
97 \( 1 - 0.730T + T^{2} \)
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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

−9.338342475984242508646113162354, −8.732026499644667138820256985785, −8.163864821765215498391822312184, −7.55443153608130529612430024418, −5.86477220051982534911339698701, −4.99127259378251016843228219584, −3.90360704639383101883266053100, −3.21017048511616586531698795348, −2.26675464875466909474560731906, −0.51790169285518240211679805072, 2.70536210234172215240043740486, 3.53543925968676485810931112374, 4.26306775247177381611223231299, 5.23202057359944672667490396942, 6.81158240698353182972480403190, 7.20950083668189539916042874385, 7.75210896912080504218105848017, 8.478729637666452095081712071167, 9.586970962451321028096970296734, 10.34053509470702453185916471506

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