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Properties

Label 2-34e2-1156.103-c0-0-0
Degree $2$
Conductor $1156$
Sign $0.781 - 0.624i$
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  + (−0.602 + 0.798i)2-s + (−0.273 − 0.961i)4-s + (1.73 − 0.673i)5-s + (0.932 + 0.361i)8-s + (0.0922 + 0.995i)9-s + (−0.510 + 1.79i)10-s + (0.172 + 0.0666i)13-s + (−0.850 + 0.526i)16-s + (−0.850 + 0.526i)17-s + (−0.850 − 0.526i)18-s + (−1.12 − 1.48i)20-s + (1.83 − 1.66i)25-s + (−0.156 + 0.0971i)26-s + (−0.0505 − 0.177i)29-s + (0.0922 − 0.995i)32-s + ⋯
L(s)  = 1  + (−0.602 + 0.798i)2-s + (−0.273 − 0.961i)4-s + (1.73 − 0.673i)5-s + (0.932 + 0.361i)8-s + (0.0922 + 0.995i)9-s + (−0.510 + 1.79i)10-s + (0.172 + 0.0666i)13-s + (−0.850 + 0.526i)16-s + (−0.850 + 0.526i)17-s + (−0.850 − 0.526i)18-s + (−1.12 − 1.48i)20-s + (1.83 − 1.66i)25-s + (−0.156 + 0.0971i)26-s + (−0.0505 − 0.177i)29-s + (0.0922 − 0.995i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.781 - 0.624i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.781 - 0.624i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.018497989\)
\(L(\frac12)\) \(\approx\) \(1.018497989\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.602 - 0.798i)T \)
17 \( 1 + (0.850 - 0.526i)T \)
good3 \( 1 + (-0.0922 - 0.995i)T^{2} \)
5 \( 1 + (-1.73 + 0.673i)T + (0.739 - 0.673i)T^{2} \)
7 \( 1 + (0.982 - 0.183i)T^{2} \)
11 \( 1 + (0.850 - 0.526i)T^{2} \)
13 \( 1 + (-0.172 - 0.0666i)T + (0.739 + 0.673i)T^{2} \)
19 \( 1 + (0.273 - 0.961i)T^{2} \)
23 \( 1 + (0.982 - 0.183i)T^{2} \)
29 \( 1 + (0.0505 + 0.177i)T + (-0.850 + 0.526i)T^{2} \)
31 \( 1 + (-0.739 - 0.673i)T^{2} \)
37 \( 1 + (0.537 + 1.07i)T + (-0.602 + 0.798i)T^{2} \)
41 \( 1 + (-1.37 - 1.25i)T + (0.0922 + 0.995i)T^{2} \)
43 \( 1 + (-0.445 - 0.895i)T^{2} \)
47 \( 1 + (0.982 + 0.183i)T^{2} \)
53 \( 1 + (0.181 + 1.95i)T + (-0.982 + 0.183i)T^{2} \)
59 \( 1 + (-0.932 - 0.361i)T^{2} \)
61 \( 1 + (1.45 - 0.271i)T + (0.932 - 0.361i)T^{2} \)
67 \( 1 + (0.273 - 0.961i)T^{2} \)
71 \( 1 + (0.982 - 0.183i)T^{2} \)
73 \( 1 + (0.156 - 0.0971i)T + (0.445 - 0.895i)T^{2} \)
79 \( 1 + (0.273 - 0.961i)T^{2} \)
83 \( 1 + (-0.0922 + 0.995i)T^{2} \)
89 \( 1 + (1.12 - 0.435i)T + (0.739 - 0.673i)T^{2} \)
97 \( 1 + (0.0505 + 0.544i)T + (-0.982 + 0.183i)T^{2} \)
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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

−9.892799718589182378220230690010, −9.194587685730624229071420237947, −8.565987097460109121066604046242, −7.72243896871272404695297503049, −6.59411310334288322142660291901, −5.97019417696875834406048012756, −5.19256625874237079512106997428, −4.48113470570963578340560204751, −2.31092646403877688716785443720, −1.52468738026870222163861167618, 1.42604584576712154741517710172, 2.49394383026765364546524919274, 3.29572481973458585869499104953, 4.60151714944977691422138597323, 5.85374769674114706976198542568, 6.61990805456316269122831702228, 7.36481273078247359207499605001, 8.752843074737745793661648803818, 9.275900294528494042917823540201, 9.846670454903261460976732275541

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