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Properties

Label 2-34e2-1156.395-c0-0-0
Degree $2$
Conductor $1156$
Sign $0.984 - 0.173i$
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  + (0.361 − 0.932i)2-s + (−0.739 − 0.673i)4-s + (0.602 + 1.79i)5-s + (−0.895 + 0.445i)8-s + (−0.961 + 0.273i)9-s + (1.89 + 0.0875i)10-s + (0.857 + 1.72i)13-s + (0.0922 + 0.995i)16-s + (−0.0922 − 0.995i)17-s + (−0.0922 + 0.995i)18-s + (0.765 − 1.73i)20-s + (−2.07 + 1.56i)25-s + (1.91 − 0.177i)26-s + (0.972 − 0.0449i)29-s + (0.961 + 0.273i)32-s + ⋯
L(s)  = 1  + (0.361 − 0.932i)2-s + (−0.739 − 0.673i)4-s + (0.602 + 1.79i)5-s + (−0.895 + 0.445i)8-s + (−0.961 + 0.273i)9-s + (1.89 + 0.0875i)10-s + (0.857 + 1.72i)13-s + (0.0922 + 0.995i)16-s + (−0.0922 − 0.995i)17-s + (−0.0922 + 0.995i)18-s + (0.765 − 1.73i)20-s + (−2.07 + 1.56i)25-s + (1.91 − 0.177i)26-s + (0.972 − 0.0449i)29-s + (0.961 + 0.273i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.165137125\)
\(L(\frac12)\) \(\approx\) \(1.165137125\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.361 + 0.932i)T \)
17 \( 1 + (0.0922 + 0.995i)T \)
good3 \( 1 + (0.961 - 0.273i)T^{2} \)
5 \( 1 + (-0.602 - 1.79i)T + (-0.798 + 0.602i)T^{2} \)
7 \( 1 + (0.526 + 0.850i)T^{2} \)
11 \( 1 + (0.995 - 0.0922i)T^{2} \)
13 \( 1 + (-0.857 - 1.72i)T + (-0.602 + 0.798i)T^{2} \)
19 \( 1 + (0.739 - 0.673i)T^{2} \)
23 \( 1 + (0.526 + 0.850i)T^{2} \)
29 \( 1 + (-0.972 + 0.0449i)T + (0.995 - 0.0922i)T^{2} \)
31 \( 1 + (-0.798 - 0.602i)T^{2} \)
37 \( 1 + (-0.666 + 0.456i)T + (0.361 - 0.932i)T^{2} \)
41 \( 1 + (-0.629 + 0.0878i)T + (0.961 - 0.273i)T^{2} \)
43 \( 1 + (-0.982 - 0.183i)T^{2} \)
47 \( 1 + (0.850 + 0.526i)T^{2} \)
53 \( 1 + (-1.01 + 0.288i)T + (0.850 - 0.526i)T^{2} \)
59 \( 1 + (0.445 + 0.895i)T^{2} \)
61 \( 1 + (1.92 + 0.453i)T + (0.895 + 0.445i)T^{2} \)
67 \( 1 + (-0.739 + 0.673i)T^{2} \)
71 \( 1 + (-0.526 - 0.850i)T^{2} \)
73 \( 1 + (1.34 + 1.11i)T + (0.183 + 0.982i)T^{2} \)
79 \( 1 + (-0.673 - 0.739i)T^{2} \)
83 \( 1 + (-0.273 + 0.961i)T^{2} \)
89 \( 1 + (-0.322 + 0.646i)T + (-0.602 - 0.798i)T^{2} \)
97 \( 1 + (-0.0806 - 0.0449i)T + (0.526 + 0.850i)T^{2} \)
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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

−10.22232444753830776683767987117, −9.392791024565690847867601212396, −8.738232000275996256517069639594, −7.38211695629183714148307354142, −6.39351020232325823476343850559, −5.95229629263879825932142524209, −4.69795286779915356644141935163, −3.54329092011308893989490745198, −2.73754649732126754482895952721, −1.95885873500417092811286789641, 0.991797458304696327724782054036, 2.96164523016712173037011917066, 4.15620726165741852692584767983, 5.04554099276788546568771763406, 5.91664166856393258897960798335, 6.06708464584799490453570440182, 7.78325367546877124688864631176, 8.416051941793474680878169126077, 8.763805020767605167417030309050, 9.648599286638949728833359242634

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