Properties

Label 2-34e2-68.47-c0-0-0
Degree $2$
Conductor $1156$
Sign $-0.615 - 0.788i$
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  + i·2-s − 4-s + (1 + i)5-s i·8-s + i·9-s + (−1 + i)10-s + 16-s − 18-s + (−1 − i)20-s + i·25-s + (−1 − i)29-s + i·32-s i·36-s + (1 + i)37-s + (1 − i)40-s + (−1 + i)41-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (1 + i)5-s i·8-s + i·9-s + (−1 + i)10-s + 16-s − 18-s + (−1 − i)20-s + i·25-s + (−1 − i)29-s + i·32-s i·36-s + (1 + i)37-s + (1 − i)40-s + (−1 + i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.615 - 0.788i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ -0.615 - 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.082126591\)
\(L(\frac12)\) \(\approx\) \(1.082126591\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
17 \( 1 \)
good3 \( 1 - iT^{2} \)
5 \( 1 + (-1 - i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (1 + i)T + iT^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + (1 - i)T - iT^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (-1 + i)T - iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-1 - i)T + iT^{2} \)
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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

−10.00970584902981600579973287175, −9.607415386083401094956271544170, −8.439657804706711083837858769858, −7.72879367723669114854597092930, −6.87669512574765925648394428030, −6.16740746409478676644620606513, −5.42975100836414362453355487544, −4.50164390692390560834591818762, −3.18467872154381935110746738526, −1.97516299636307899317861391759, 1.06916387095544160553343369755, 2.13342109937984989800887388856, 3.41317344429636947129541897105, 4.38084617078799069393105239350, 5.37946989256413027767597702369, 5.97990528087817090924620174529, 7.29810120254522126109350699028, 8.595052282691261720801964921958, 9.062988826822614822111352980273, 9.652845597817256245731247013466

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