Properties

Label 2-34e2-1156.727-c0-0-0
Degree $2$
Conductor $1156$
Sign $-0.460 + 0.887i$
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  + (0.526 − 0.850i)2-s + (−0.445 − 0.895i)4-s + (0.982 − 1.18i)5-s + (−0.995 − 0.0922i)8-s + (−0.361 + 0.932i)9-s + (−0.488 − 1.45i)10-s + (0.0666 − 0.719i)13-s + (−0.602 + 0.798i)16-s + (0.602 − 0.798i)17-s + (0.602 + 0.798i)18-s + (−1.49 − 0.352i)20-s + (−0.251 − 1.34i)25-s + (−0.576 − 0.435i)26-s + (−0.581 + 1.73i)29-s + (0.361 + 0.932i)32-s + ⋯
L(s)  = 1  + (0.526 − 0.850i)2-s + (−0.445 − 0.895i)4-s + (0.982 − 1.18i)5-s + (−0.995 − 0.0922i)8-s + (−0.361 + 0.932i)9-s + (−0.488 − 1.45i)10-s + (0.0666 − 0.719i)13-s + (−0.602 + 0.798i)16-s + (0.602 − 0.798i)17-s + (0.602 + 0.798i)18-s + (−1.49 − 0.352i)20-s + (−0.251 − 1.34i)25-s + (−0.576 − 0.435i)26-s + (−0.581 + 1.73i)29-s + (0.361 + 0.932i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.460 + 0.887i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ -0.460 + 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.440863887\)
\(L(\frac12)\) \(\approx\) \(1.440863887\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.526 + 0.850i)T \)
17 \( 1 + (-0.602 + 0.798i)T \)
good3 \( 1 + (0.361 - 0.932i)T^{2} \)
5 \( 1 + (-0.982 + 1.18i)T + (-0.183 - 0.982i)T^{2} \)
7 \( 1 + (-0.673 + 0.739i)T^{2} \)
11 \( 1 + (-0.798 - 0.602i)T^{2} \)
13 \( 1 + (-0.0666 + 0.719i)T + (-0.982 - 0.183i)T^{2} \)
19 \( 1 + (0.445 - 0.895i)T^{2} \)
23 \( 1 + (-0.673 + 0.739i)T^{2} \)
29 \( 1 + (0.581 - 1.73i)T + (-0.798 - 0.602i)T^{2} \)
31 \( 1 + (-0.183 + 0.982i)T^{2} \)
37 \( 1 + (1.70 - 0.947i)T + (0.526 - 0.850i)T^{2} \)
41 \( 1 + (-1.05 + 0.722i)T + (0.361 - 0.932i)T^{2} \)
43 \( 1 + (-0.273 + 0.961i)T^{2} \)
47 \( 1 + (-0.739 + 0.673i)T^{2} \)
53 \( 1 + (0.486 - 1.25i)T + (-0.739 - 0.673i)T^{2} \)
59 \( 1 + (0.0922 - 0.995i)T^{2} \)
61 \( 1 + (-1.64 + 0.0762i)T + (0.995 - 0.0922i)T^{2} \)
67 \( 1 + (-0.445 + 0.895i)T^{2} \)
71 \( 1 + (0.673 - 0.739i)T^{2} \)
73 \( 1 + (0.800 - 0.111i)T + (0.961 - 0.273i)T^{2} \)
79 \( 1 + (0.895 + 0.445i)T^{2} \)
83 \( 1 + (0.932 - 0.361i)T^{2} \)
89 \( 1 + (-0.0971 - 1.04i)T + (-0.982 + 0.183i)T^{2} \)
97 \( 1 + (0.765 + 1.73i)T + (-0.673 + 0.739i)T^{2} \)
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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

−9.896490513618054923832820375327, −8.992781653439499241162647361684, −8.491853356522543088448658587593, −7.21674057894720337812879834755, −5.79975697393362416254235905403, −5.30844483381953853917919128055, −4.77234444345632945432865716309, −3.37660097642310406507842978312, −2.28053024422713153202601316605, −1.22589710632426478956440652909, 2.21412855389388690522458511680, 3.31972999709667839033872111224, 4.11184494764402618009058260278, 5.57177199574147810716045109470, 6.10329128058874121551779158511, 6.71949215788156549927949681624, 7.51801635897850325534016108338, 8.574090953704376417196168085881, 9.426835984947397577002788627525, 10.04039608320705188061782303285

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