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Properties

Label 2-34e2-1156.183-c0-0-0
Degree $2$
Conductor $1156$
Sign $0.984 - 0.173i$
Analytic cond. $0.576919$
Root an. cond. $0.759551$
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  + (−0.361 + 0.932i)2-s + (−0.739 − 0.673i)4-s + (0.602 − 0.201i)5-s + (0.895 − 0.445i)8-s + (0.961 − 0.273i)9-s + (−0.0293 + 0.634i)10-s + (−0.857 − 1.72i)13-s + (0.0922 + 0.995i)16-s + (−0.0922 − 0.995i)17-s + (−0.0922 + 0.995i)18-s + (−0.581 − 0.256i)20-s + (−0.475 + 0.359i)25-s + (1.91 − 0.177i)26-s + (−0.0806 − 1.74i)29-s + (−0.961 − 0.273i)32-s + ⋯
L(s)  = 1  + (−0.361 + 0.932i)2-s + (−0.739 − 0.673i)4-s + (0.602 − 0.201i)5-s + (0.895 − 0.445i)8-s + (0.961 − 0.273i)9-s + (−0.0293 + 0.634i)10-s + (−0.857 − 1.72i)13-s + (0.0922 + 0.995i)16-s + (−0.0922 − 0.995i)17-s + (−0.0922 + 0.995i)18-s + (−0.581 − 0.256i)20-s + (−0.475 + 0.359i)25-s + (1.91 − 0.177i)26-s + (−0.0806 − 1.74i)29-s + (−0.961 − 0.273i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9197685336\)
\(L(\frac12)\) \(\approx\) \(0.9197685336\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.361 - 0.932i)T \)
17 \( 1 + (0.0922 + 0.995i)T \)
good3 \( 1 + (-0.961 + 0.273i)T^{2} \)
5 \( 1 + (-0.602 + 0.201i)T + (0.798 - 0.602i)T^{2} \)
7 \( 1 + (-0.526 - 0.850i)T^{2} \)
11 \( 1 + (-0.995 + 0.0922i)T^{2} \)
13 \( 1 + (0.857 + 1.72i)T + (-0.602 + 0.798i)T^{2} \)
19 \( 1 + (0.739 - 0.673i)T^{2} \)
23 \( 1 + (-0.526 - 0.850i)T^{2} \)
29 \( 1 + (0.0806 + 1.74i)T + (-0.995 + 0.0922i)T^{2} \)
31 \( 1 + (0.798 + 0.602i)T^{2} \)
37 \( 1 + (-1.03 - 1.50i)T + (-0.361 + 0.932i)T^{2} \)
41 \( 1 + (-0.261 - 1.87i)T + (-0.961 + 0.273i)T^{2} \)
43 \( 1 + (-0.982 - 0.183i)T^{2} \)
47 \( 1 + (0.850 + 0.526i)T^{2} \)
53 \( 1 + (-1.01 + 0.288i)T + (0.850 - 0.526i)T^{2} \)
59 \( 1 + (0.445 + 0.895i)T^{2} \)
61 \( 1 + (-0.0632 + 0.268i)T + (-0.895 - 0.445i)T^{2} \)
67 \( 1 + (-0.739 + 0.673i)T^{2} \)
71 \( 1 + (0.526 + 0.850i)T^{2} \)
73 \( 1 + (0.621 - 0.748i)T + (-0.183 - 0.982i)T^{2} \)
79 \( 1 + (0.673 + 0.739i)T^{2} \)
83 \( 1 + (-0.273 + 0.961i)T^{2} \)
89 \( 1 + (0.322 - 0.646i)T + (-0.602 - 0.798i)T^{2} \)
97 \( 1 + (0.972 - 1.74i)T + (-0.526 - 0.850i)T^{2} \)
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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

−9.750288350636213355147447398914, −9.444970043131218050379687623144, −8.081264287973319655858191852282, −7.67988861065304234486171439290, −6.72054602224783022737367872870, −5.85702067153212860526988467595, −5.08873218546387182189107431717, −4.24354414806437519497179386790, −2.68367042995652432227295156658, −1.03961452497543225094037625698, 1.70256817407606003915508432947, 2.30274429549523576517097640186, 3.85487751082992154827027934823, 4.46667691419254689796873967715, 5.60912235733355310803437824609, 6.92517106105783824522100481215, 7.44081583645198373238091395878, 8.714295132044060826279476300473, 9.270999219118082148837208968264, 10.07483666336139410005195093612

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