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Properties

Label 2-34e2-1156.1087-c0-0-0
Degree $2$
Conductor $1156$
Sign $0.947 - 0.320i$
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  + (−0.0922 − 0.995i)2-s + (−0.982 + 0.183i)4-s + (−1.85 − 0.526i)5-s + (0.273 + 0.961i)8-s + (−0.445 + 0.895i)9-s + (−0.353 + 1.89i)10-s + (0.243 + 0.857i)13-s + (0.932 − 0.361i)16-s + (0.932 − 0.361i)17-s + (0.932 + 0.361i)18-s + (1.91 + 0.177i)20-s + (2.29 + 1.42i)25-s + (0.831 − 0.322i)26-s + (0.328 + 1.75i)29-s + (−0.445 − 0.895i)32-s + ⋯
L(s)  = 1  + (−0.0922 − 0.995i)2-s + (−0.982 + 0.183i)4-s + (−1.85 − 0.526i)5-s + (0.273 + 0.961i)8-s + (−0.445 + 0.895i)9-s + (−0.353 + 1.89i)10-s + (0.243 + 0.857i)13-s + (0.932 − 0.361i)16-s + (0.932 − 0.361i)17-s + (0.932 + 0.361i)18-s + (1.91 + 0.177i)20-s + (2.29 + 1.42i)25-s + (0.831 − 0.322i)26-s + (0.328 + 1.75i)29-s + (−0.445 − 0.895i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.947 - 0.320i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (1087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.947 - 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4661969526\)
\(L(\frac12)\) \(\approx\) \(0.4661969526\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0922 + 0.995i)T \)
17 \( 1 + (-0.932 + 0.361i)T \)
good3 \( 1 + (0.445 - 0.895i)T^{2} \)
5 \( 1 + (1.85 + 0.526i)T + (0.850 + 0.526i)T^{2} \)
7 \( 1 + (-0.602 - 0.798i)T^{2} \)
11 \( 1 + (0.932 - 0.361i)T^{2} \)
13 \( 1 + (-0.243 - 0.857i)T + (-0.850 + 0.526i)T^{2} \)
19 \( 1 + (0.982 + 0.183i)T^{2} \)
23 \( 1 + (-0.602 - 0.798i)T^{2} \)
29 \( 1 + (-0.328 - 1.75i)T + (-0.932 + 0.361i)T^{2} \)
31 \( 1 + (-0.850 + 0.526i)T^{2} \)
37 \( 1 + (1.34 - 1.47i)T + (-0.0922 - 0.995i)T^{2} \)
41 \( 1 + (1.01 + 1.63i)T + (-0.445 + 0.895i)T^{2} \)
43 \( 1 + (-0.739 - 0.673i)T^{2} \)
47 \( 1 + (0.602 - 0.798i)T^{2} \)
53 \( 1 + (0.537 - 1.07i)T + (-0.602 - 0.798i)T^{2} \)
59 \( 1 + (0.273 + 0.961i)T^{2} \)
61 \( 1 + (0.840 - 0.634i)T + (0.273 - 0.961i)T^{2} \)
67 \( 1 + (0.982 + 0.183i)T^{2} \)
71 \( 1 + (-0.602 - 0.798i)T^{2} \)
73 \( 1 + (-0.646 - 1.66i)T + (-0.739 + 0.673i)T^{2} \)
79 \( 1 + (-0.982 - 0.183i)T^{2} \)
83 \( 1 + (-0.445 - 0.895i)T^{2} \)
89 \( 1 + (-0.0505 + 0.177i)T + (-0.850 - 0.526i)T^{2} \)
97 \( 1 + (-0.328 - 0.163i)T + (0.602 + 0.798i)T^{2} \)
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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

−10.32702603108617945715269490342, −8.965349875021027035083787038466, −8.613324336420931144317678695722, −7.79638482414143238962295337206, −7.07969855168377677610949515370, −5.28512380411110578804626344642, −4.70739435954526970234842868267, −3.75530452242500841011224012101, −3.00326244258350917579313255972, −1.37454153953959900617315380185, 0.48680214664612358280771602990, 3.31193864998621159621082374776, 3.71927521467983542560920003063, 4.81539506095129577271570619713, 5.94998952808299838548447543368, 6.70828272387722149743137454624, 7.63166794000915641726973520868, 8.089490342095046976836439803237, 8.753288147983193481833107171378, 9.873246171587216475576477074977

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