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Properties

Label 2-34e2-1156.815-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 + (−0.0675 − 0.361i)5-s + (0.982 + 0.183i)8-s + (−0.739 − 0.673i)9-s + (−0.293 + 0.221i)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 + (0.328 + 0.163i)20-s + (0.806 − 0.312i)25-s + (−0.404 − 1.42i)26-s + (−1.07 − 0.811i)29-s + (−0.739 + 0.673i)32-s + ⋯
L(s)  = 1  + (−0.445 − 0.895i)2-s + (−0.602 + 0.798i)4-s + (−0.0675 − 0.361i)5-s + (0.982 + 0.183i)8-s + (−0.739 − 0.673i)9-s + (−0.293 + 0.221i)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 + (0.328 + 0.163i)20-s + (0.806 − 0.312i)25-s + (−0.404 − 1.42i)26-s + (−1.07 − 0.811i)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: $\chi_{1156} (815, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ -0.304 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7437383091\)
\(L(\frac12)\) \(\approx\) \(0.7437383091\)
\(L(1)\) 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 \( 1 + (0.0675 + 0.361i)T + (-0.932 + 0.361i)T^{2} \)
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 \( 1 + (1.07 + 0.811i)T + (0.273 + 0.961i)T^{2} \)
31 \( 1 + (0.932 + 0.361i)T^{2} \)
37 \( 1 + (-0.942 + 1.52i)T + (-0.445 - 0.895i)T^{2} \)
41 \( 1 + (0.132 - 0.342i)T + (-0.739 - 0.673i)T^{2} \)
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 \( 1 + (-0.719 + 0.0666i)T + (0.982 - 0.183i)T^{2} \)
67 \( 1 + (0.602 + 0.798i)T^{2} \)
71 \( 1 + (0.0922 + 0.995i)T^{2} \)
73 \( 1 + (1.29 - 0.368i)T + (0.850 - 0.526i)T^{2} \)
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 \( 1 + (1.07 - 1.17i)T + (-0.0922 - 0.995i)T^{2} \)
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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

−9.599439401611446562946847485898, −8.996459950744848023589623834626, −8.479593979251309268538484421773, −7.50985420181126410671087712954, −6.41234623871761906464567325556, −5.39410537961781895127148151667, −4.22307902140072231657030760072, −3.45038202231710739394688995545, −2.33714406050644589714499765520, −0.855291356762008806064445845852, 1.54470289932736364512309431062, 3.16800200816067027092805909176, 4.33432146548707859826467128359, 5.45847681471205741733626216610, 6.08998258489866508943868447919, 6.91036943593123172682690537755, 7.908849449310260379815051199243, 8.489825754481598937164020805171, 9.123337566672115289621838648355, 10.26897729131142526904132805010

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