Properties

Label 2-34e2-1156.1143-c0-0-0
Degree $2$
Conductor $1156$
Sign $-0.712 - 0.701i$
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  + (−0.961 + 0.273i)2-s + (0.850 − 0.526i)4-s + (−0.0922 + 1.99i)5-s + (−0.673 + 0.739i)8-s + (0.183 + 0.982i)9-s + (−0.457 − 1.94i)10-s + (−0.271 − 0.247i)13-s + (0.445 − 0.895i)16-s + (−0.445 + 0.895i)17-s + (−0.445 − 0.895i)18-s + (0.972 + 1.74i)20-s + (−2.97 − 0.276i)25-s + (0.328 + 0.163i)26-s + (0.377 − 1.60i)29-s + (−0.183 + 0.982i)32-s + ⋯
L(s)  = 1  + (−0.961 + 0.273i)2-s + (0.850 − 0.526i)4-s + (−0.0922 + 1.99i)5-s + (−0.673 + 0.739i)8-s + (0.183 + 0.982i)9-s + (−0.457 − 1.94i)10-s + (−0.271 − 0.247i)13-s + (0.445 − 0.895i)16-s + (−0.445 + 0.895i)17-s + (−0.445 − 0.895i)18-s + (0.972 + 1.74i)20-s + (−2.97 − 0.276i)25-s + (0.328 + 0.163i)26-s + (0.377 − 1.60i)29-s + (−0.183 + 0.982i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.712 - 0.701i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (1143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ -0.712 - 0.701i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5910302806\)
\(L(\frac12)\) \(\approx\) \(0.5910302806\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.961 - 0.273i)T \)
17 \( 1 + (0.445 - 0.895i)T \)
good3 \( 1 + (-0.183 - 0.982i)T^{2} \)
5 \( 1 + (0.0922 - 1.99i)T + (-0.995 - 0.0922i)T^{2} \)
7 \( 1 + (-0.361 - 0.932i)T^{2} \)
11 \( 1 + (-0.895 - 0.445i)T^{2} \)
13 \( 1 + (0.271 + 0.247i)T + (0.0922 + 0.995i)T^{2} \)
19 \( 1 + (-0.850 - 0.526i)T^{2} \)
23 \( 1 + (-0.361 - 0.932i)T^{2} \)
29 \( 1 + (-0.377 + 1.60i)T + (-0.895 - 0.445i)T^{2} \)
31 \( 1 + (-0.995 + 0.0922i)T^{2} \)
37 \( 1 + (0.134 + 0.963i)T + (-0.961 + 0.273i)T^{2} \)
41 \( 1 + (0.0590 - 0.0710i)T + (-0.183 - 0.982i)T^{2} \)
43 \( 1 + (-0.602 + 0.798i)T^{2} \)
47 \( 1 + (-0.932 - 0.361i)T^{2} \)
53 \( 1 + (-0.132 - 0.710i)T + (-0.932 + 0.361i)T^{2} \)
59 \( 1 + (0.739 + 0.673i)T^{2} \)
61 \( 1 + (-1.16 - 0.516i)T + (0.673 + 0.739i)T^{2} \)
67 \( 1 + (0.850 + 0.526i)T^{2} \)
71 \( 1 + (0.361 + 0.932i)T^{2} \)
73 \( 1 + (-0.359 - 1.07i)T + (-0.798 + 0.602i)T^{2} \)
79 \( 1 + (-0.526 + 0.850i)T^{2} \)
83 \( 1 + (-0.982 - 0.183i)T^{2} \)
89 \( 1 + (1.42 - 1.29i)T + (0.0922 - 0.995i)T^{2} \)
97 \( 1 + (1.10 - 1.60i)T + (-0.361 - 0.932i)T^{2} \)
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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

−10.19654326819589028865588584782, −9.813329249594524013781989533644, −8.475746569550957035104538016736, −7.70686198476924681754422369795, −7.18969871504468193523925094406, −6.35173046435067563450922290202, −5.65827885123246235803309280798, −4.06195180476313779918231823928, −2.73407228026501518191925889141, −2.11235309736635328512070535690, 0.71297059867034199720794574493, 1.79857956654399455668092766484, 3.37656283740855865314063876012, 4.47121627280449241303164359610, 5.35557179032733201960467522213, 6.55215079085436351405773944098, 7.38814475387878325932612616539, 8.478515367490402471691811393708, 8.840728682437282077512493371611, 9.511298904193538308274607214665

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