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Properties

Label 2-34e2-1156.203-c0-0-0
Degree $2$
Conductor $1156$
Sign $0.701 + 0.712i$
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  + (0.273 − 0.961i)2-s + (−0.850 − 0.526i)4-s + (−0.907 + 0.995i)5-s + (−0.739 + 0.673i)8-s + (0.982 + 0.183i)9-s + (0.709 + 1.14i)10-s + (1.45 − 1.32i)13-s + (0.445 + 0.895i)16-s + (0.445 + 0.895i)17-s + (0.445 − 0.895i)18-s + (1.29 − 0.368i)20-s + (−0.0752 − 0.811i)25-s + (−0.876 − 1.75i)26-s + (−0.193 + 0.312i)29-s + (0.982 − 0.183i)32-s + ⋯
L(s)  = 1  + (0.273 − 0.961i)2-s + (−0.850 − 0.526i)4-s + (−0.907 + 0.995i)5-s + (−0.739 + 0.673i)8-s + (0.982 + 0.183i)9-s + (0.709 + 1.14i)10-s + (1.45 − 1.32i)13-s + (0.445 + 0.895i)16-s + (0.445 + 0.895i)17-s + (0.445 − 0.895i)18-s + (1.29 − 0.368i)20-s + (−0.0752 − 0.811i)25-s + (−0.876 − 1.75i)26-s + (−0.193 + 0.312i)29-s + (0.982 − 0.183i)32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.701 + 0.712i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.052149648\)
\(L(\frac12)\) \(\approx\) \(1.052149648\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.273 + 0.961i)T \)
17 \( 1 + (-0.445 - 0.895i)T \)
good3 \( 1 + (-0.982 - 0.183i)T^{2} \)
5 \( 1 + (0.907 - 0.995i)T + (-0.0922 - 0.995i)T^{2} \)
7 \( 1 + (0.932 + 0.361i)T^{2} \)
11 \( 1 + (0.445 + 0.895i)T^{2} \)
13 \( 1 + (-1.45 + 1.32i)T + (0.0922 - 0.995i)T^{2} \)
19 \( 1 + (0.850 - 0.526i)T^{2} \)
23 \( 1 + (0.932 + 0.361i)T^{2} \)
29 \( 1 + (0.193 - 0.312i)T + (-0.445 - 0.895i)T^{2} \)
31 \( 1 + (0.0922 - 0.995i)T^{2} \)
37 \( 1 + (-1.53 + 1.15i)T + (0.273 - 0.961i)T^{2} \)
41 \( 1 + (-1.34 - 0.124i)T + (0.982 + 0.183i)T^{2} \)
43 \( 1 + (0.602 + 0.798i)T^{2} \)
47 \( 1 + (-0.932 + 0.361i)T^{2} \)
53 \( 1 + (1.83 + 0.342i)T + (0.932 + 0.361i)T^{2} \)
59 \( 1 + (-0.739 + 0.673i)T^{2} \)
61 \( 1 + (0.719 - 1.85i)T + (-0.739 - 0.673i)T^{2} \)
67 \( 1 + (0.850 - 0.526i)T^{2} \)
71 \( 1 + (0.932 + 0.361i)T^{2} \)
73 \( 1 + (0.328 - 0.163i)T + (0.602 - 0.798i)T^{2} \)
79 \( 1 + (-0.850 + 0.526i)T^{2} \)
83 \( 1 + (0.982 - 0.183i)T^{2} \)
89 \( 1 + (-0.404 - 0.368i)T + (0.0922 + 0.995i)T^{2} \)
97 \( 1 + (0.193 - 1.03i)T + (-0.932 - 0.361i)T^{2} \)
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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

−10.23151467611794046620350357169, −9.297342454502210641804252666874, −8.128671516944279732088370523676, −7.67895382535430247908202225206, −6.40697761167907435606660948107, −5.60404846513164993326201516111, −4.25126074267269618165755246343, −3.66086632386963976420986757119, −2.83338690964142211479623405222, −1.29975557571345099482002491333, 1.18408560457446018734734438584, 3.41870578512578101064778384167, 4.37391512214336856915205362607, 4.67267238947128959082283325837, 6.02882843987016664324519371206, 6.74612184402807536765117291850, 7.73783147409510055098761743305, 8.209342535383463263693337673970, 9.280091241294008570554254280553, 9.519480735128088465321731150082

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