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Properties

Label 2-1176-168.53-c0-0-4
Degree $2$
Conductor $1176$
Sign $-0.991 + 0.126i$
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  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 0.999·6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)12-s − 0.999·15-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)18-s + 0.999·20-s − 0.999·22-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 0.999·6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)12-s − 0.999·15-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)18-s + 0.999·20-s − 0.999·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ -0.991 + 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7762427453\)
\(L(\frac12)\) \(\approx\) \(0.7762427453\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.185071813372851386745157497706, −8.907037776560684756546142192240, −8.123674128320539566041894563446, −7.48543079372627703802208122715, −6.42905678795096232028728501384, −5.16674276485991631866735976195, −3.95038864210435977563621553114, −3.22029161399103696854180876123, −1.92269843499679796701520053419, −0.796102723187690449342926660761, 2.10145150397294409097414992252, 3.53830671774421151814635561205, 4.33627918375669845845352337523, 5.30194883798291579962600069860, 6.34137612611786791548647542959, 7.32380998864990173117437317453, 7.75414314137893028240538963046, 8.820130259655560051029766663259, 9.460920323510715531673400701629, 10.11649467267522829734197393932

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