Properties

Label 2-34e2-1156.919-c0-0-0
Degree $2$
Conductor $1156$
Sign $0.304 - 0.952i$
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  + (0.445 + 0.895i)2-s + (−0.602 + 0.798i)4-s + (1.93 − 0.361i)5-s + (−0.982 − 0.183i)8-s + (0.739 + 0.673i)9-s + (1.18 + 1.56i)10-s + (−1.45 − 0.271i)13-s + (−0.273 − 0.961i)16-s + (−0.273 − 0.961i)17-s + (−0.273 + 0.961i)18-s + (−0.876 + 1.75i)20-s + (2.67 − 1.03i)25-s + (−0.404 − 1.42i)26-s + (−0.890 + 1.17i)29-s + (0.739 − 0.673i)32-s + ⋯
L(s)  = 1  + (0.445 + 0.895i)2-s + (−0.602 + 0.798i)4-s + (1.93 − 0.361i)5-s + (−0.982 − 0.183i)8-s + (0.739 + 0.673i)9-s + (1.18 + 1.56i)10-s + (−1.45 − 0.271i)13-s + (−0.273 − 0.961i)16-s + (−0.273 − 0.961i)17-s + (−0.273 + 0.961i)18-s + (−0.876 + 1.75i)20-s + (2.67 − 1.03i)25-s + (−0.404 − 1.42i)26-s + (−0.890 + 1.17i)29-s + (0.739 − 0.673i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.304 - 0.952i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (919, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.304 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.600242493\)
\(L(\frac12)\) \(\approx\) \(1.600242493\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.445 - 0.895i)T \)
17 \( 1 + (0.273 + 0.961i)T \)
good3 \( 1 + (-0.739 - 0.673i)T^{2} \)
5 \( 1 + (-1.93 + 0.361i)T + (0.932 - 0.361i)T^{2} \)
7 \( 1 + (-0.0922 - 0.995i)T^{2} \)
11 \( 1 + (0.273 + 0.961i)T^{2} \)
13 \( 1 + (1.45 + 0.271i)T + (0.932 + 0.361i)T^{2} \)
19 \( 1 + (0.602 + 0.798i)T^{2} \)
23 \( 1 + (-0.0922 - 0.995i)T^{2} \)
29 \( 1 + (0.890 - 1.17i)T + (-0.273 - 0.961i)T^{2} \)
31 \( 1 + (-0.932 - 0.361i)T^{2} \)
37 \( 1 + (0.757 + 0.469i)T + (0.445 + 0.895i)T^{2} \)
41 \( 1 + (1.83 + 0.710i)T + (0.739 + 0.673i)T^{2} \)
43 \( 1 + (0.850 + 0.526i)T^{2} \)
47 \( 1 + (-0.0922 + 0.995i)T^{2} \)
53 \( 1 + (-0.136 - 0.124i)T + (0.0922 + 0.995i)T^{2} \)
59 \( 1 + (0.982 + 0.183i)T^{2} \)
61 \( 1 + (-0.172 - 1.85i)T + (-0.982 + 0.183i)T^{2} \)
67 \( 1 + (0.602 + 0.798i)T^{2} \)
71 \( 1 + (-0.0922 - 0.995i)T^{2} \)
73 \( 1 + (0.404 + 1.42i)T + (-0.850 + 0.526i)T^{2} \)
79 \( 1 + (0.602 + 0.798i)T^{2} \)
83 \( 1 + (-0.739 + 0.673i)T^{2} \)
89 \( 1 + (0.876 - 0.163i)T + (0.932 - 0.361i)T^{2} \)
97 \( 1 + (0.890 + 0.811i)T + (0.0922 + 0.995i)T^{2} \)
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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

−9.916589352822742420383789659449, −9.321553051667462363877920548427, −8.604281509717314028999239275185, −7.26924744925333315087420438910, −6.97120557832501229053591965582, −5.73789840424336808113184440999, −5.17889292700883542755905128143, −4.60317115445292823801033797380, −2.90764237320056273986029850714, −1.88886315088026377400314852111, 1.65715191141002781406647668283, 2.28110510539716804046810084134, 3.46564918610721445163387014426, 4.68552091801567158919153803852, 5.47197271719074349378145596950, 6.35249290787442743983206304397, 6.95403178762972784756967012829, 8.567917414102033230034570677089, 9.574992826529639529705125653143, 9.851406553907865463510111650600

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