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Properties

Label 2-34e2-1156.135-c0-0-0
Degree $2$
Conductor $1156$
Sign $0.960 + 0.278i$
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  + (0.982 + 0.183i)2-s + (0.932 + 0.361i)4-s + (−0.554 − 0.895i)5-s + (0.850 + 0.526i)8-s + (0.602 − 0.798i)9-s + (−0.380 − 0.981i)10-s + (−1.02 − 0.634i)13-s + (0.739 + 0.673i)16-s + (0.739 + 0.673i)17-s + (0.739 − 0.673i)18-s + (−0.193 − 1.03i)20-s + (−0.0483 + 0.0971i)25-s + (−0.890 − 0.811i)26-s + (−0.576 + 1.48i)29-s + (0.602 + 0.798i)32-s + ⋯
L(s)  = 1  + (0.982 + 0.183i)2-s + (0.932 + 0.361i)4-s + (−0.554 − 0.895i)5-s + (0.850 + 0.526i)8-s + (0.602 − 0.798i)9-s + (−0.380 − 0.981i)10-s + (−1.02 − 0.634i)13-s + (0.739 + 0.673i)16-s + (0.739 + 0.673i)17-s + (0.739 − 0.673i)18-s + (−0.193 − 1.03i)20-s + (−0.0483 + 0.0971i)25-s + (−0.890 − 0.811i)26-s + (−0.576 + 1.48i)29-s + (0.602 + 0.798i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.960 + 0.278i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.960 + 0.278i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.794971607\)
\(L(\frac12)\) \(\approx\) \(1.794971607\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.982 - 0.183i)T \)
17 \( 1 + (-0.739 - 0.673i)T \)
good3 \( 1 + (-0.602 + 0.798i)T^{2} \)
5 \( 1 + (0.554 + 0.895i)T + (-0.445 + 0.895i)T^{2} \)
7 \( 1 + (-0.273 - 0.961i)T^{2} \)
11 \( 1 + (0.739 + 0.673i)T^{2} \)
13 \( 1 + (1.02 + 0.634i)T + (0.445 + 0.895i)T^{2} \)
19 \( 1 + (-0.932 + 0.361i)T^{2} \)
23 \( 1 + (-0.273 - 0.961i)T^{2} \)
29 \( 1 + (0.576 - 1.48i)T + (-0.739 - 0.673i)T^{2} \)
31 \( 1 + (0.445 + 0.895i)T^{2} \)
37 \( 1 + (0.365 + 0.0339i)T + (0.982 + 0.183i)T^{2} \)
41 \( 1 + (0.942 - 0.469i)T + (0.602 - 0.798i)T^{2} \)
43 \( 1 + (-0.0922 + 0.995i)T^{2} \)
47 \( 1 + (0.273 - 0.961i)T^{2} \)
53 \( 1 + (-0.329 + 0.436i)T + (-0.273 - 0.961i)T^{2} \)
59 \( 1 + (0.850 + 0.526i)T^{2} \)
61 \( 1 + (1.72 - 0.489i)T + (0.850 - 0.526i)T^{2} \)
67 \( 1 + (-0.932 + 0.361i)T^{2} \)
71 \( 1 + (-0.273 - 0.961i)T^{2} \)
73 \( 1 + (-1.07 + 1.17i)T + (-0.0922 - 0.995i)T^{2} \)
79 \( 1 + (0.932 - 0.361i)T^{2} \)
83 \( 1 + (0.602 + 0.798i)T^{2} \)
89 \( 1 + (1.67 - 1.03i)T + (0.445 - 0.895i)T^{2} \)
97 \( 1 + (0.576 + 0.435i)T + (0.273 + 0.961i)T^{2} \)
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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

−10.06706295507933575298841500907, −9.054358931473362507320195822207, −8.089829924418128477321066536574, −7.42326737145714415271847980229, −6.56716115062235479634095840296, −5.49955892845792557878241895539, −4.80192745306717756972990534308, −3.92454956943859865693968939851, −3.06433417513601149814297403798, −1.45058883492430517636768617904, 1.95925444809955159535992324438, 2.90576493237275265068458125636, 3.93015085882034604302827768668, 4.77396694661198314396230551195, 5.63595430999851957833209812169, 6.87315530382093505474111182115, 7.26682839154861595135534226752, 7.997854580358814393410188040471, 9.579716829864106219208320699949, 10.19488023814903711866728805278

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