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Properties

Label 2-34e2-1156.715-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 + (1.44 + 0.895i)5-s + (−0.850 + 0.526i)8-s + (−0.602 − 0.798i)9-s + (−1.58 − 0.614i)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 + (1.67 + 0.312i)20-s + (0.843 + 1.69i)25-s + (−0.890 + 0.811i)26-s + (−1.12 + 0.435i)29-s + (−0.602 + 0.798i)32-s + ⋯
L(s)  = 1  + (−0.982 + 0.183i)2-s + (0.932 − 0.361i)4-s + (1.44 + 0.895i)5-s + (−0.850 + 0.526i)8-s + (−0.602 − 0.798i)9-s + (−1.58 − 0.614i)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 + (1.67 + 0.312i)20-s + (0.843 + 1.69i)25-s + (−0.890 + 0.811i)26-s + (−1.12 + 0.435i)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: $\chi_{1156} (715, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.960 - 0.278i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8868926366\)
\(L(\frac12)\) \(\approx\) \(0.8868926366\)
\(L(1)\) 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 \( 1 + (-1.44 - 0.895i)T + (0.445 + 0.895i)T^{2} \)
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 \( 1 + (1.12 - 0.435i)T + (0.739 - 0.673i)T^{2} \)
31 \( 1 + (-0.445 + 0.895i)T^{2} \)
37 \( 1 + (0.181 + 1.95i)T + (-0.982 + 0.183i)T^{2} \)
41 \( 1 + (0.757 - 1.52i)T + (-0.602 - 0.798i)T^{2} \)
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 \( 1 + (0.243 - 0.857i)T + (-0.850 - 0.526i)T^{2} \)
67 \( 1 + (-0.932 - 0.361i)T^{2} \)
71 \( 1 + (0.273 - 0.961i)T^{2} \)
73 \( 1 + (0.890 - 0.811i)T + (0.0922 - 0.995i)T^{2} \)
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 \( 1 + (1.12 + 1.48i)T + (-0.273 + 0.961i)T^{2} \)
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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

−9.823384165498571305289542991461, −9.362155409777064657740152984281, −8.595314802880588991379961605293, −7.51453701339323676305789563996, −6.71872967980270274005034119795, −5.86141878869311858834922823150, −5.58706138948335173600626234714, −3.39135543892844762152711327218, −2.64530173180406824203638220628, −1.36594786777538793152864852565, 1.46638673108833553140581375472, 2.14707144499080677072344353536, 3.54673258559454938634182342656, 5.08199126451553374220240497868, 5.87309969674189382664105234944, 6.53458770415991441791066771063, 7.81199769355990484380387806577, 8.602179892857686563625334967388, 9.020994395308215322438938889586, 9.965317745622446321800695368763

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