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Properties

Label 2-1183-91.46-c0-0-0
Degree $2$
Conductor $1183$
Sign $0.934 + 0.355i$
Analytic cond. $0.590393$
Root an. cond. $0.768370$
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  + (0.707 + 0.707i)2-s + (−0.5 − 0.866i)3-s + (−0.258 + 0.965i)5-s + (0.258 − 0.965i)6-s + (−0.965 − 0.258i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.500i)10-s + (0.965 + 0.258i)11-s + (−0.500 − 0.866i)14-s + (0.965 − 0.258i)15-s + 1.00·16-s − i·17-s + (0.965 − 0.258i)19-s + (0.258 + 0.965i)21-s + (0.500 + 0.866i)22-s − i·23-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.5 − 0.866i)3-s + (−0.258 + 0.965i)5-s + (0.258 − 0.965i)6-s + (−0.965 − 0.258i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.500i)10-s + (0.965 + 0.258i)11-s + (−0.500 − 0.866i)14-s + (0.965 − 0.258i)15-s + 1.00·16-s − i·17-s + (0.965 − 0.258i)19-s + (0.258 + 0.965i)21-s + (0.500 + 0.866i)22-s − i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.934 + 0.355i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ 0.934 + 0.355i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.240302346\)
\(L(\frac12)\) \(\approx\) \(1.240302346\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.965 + 0.258i)T \)
13 \( 1 \)
good2 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
3 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + iT - T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
41 \( 1 + (0.866 - 0.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T^{2} \)
73 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
97 \( 1 + (0.517 - 1.93i)T + (-0.866 - 0.5i)T^{2} \)
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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

−9.917674087550959424844142437591, −9.251684879068943421773767449104, −7.68355233922151571108385941477, −6.97812960000007871181808720226, −6.64658149058915866825316501038, −6.11065844596119734346392268135, −4.94371581211301600581684016243, −3.87531513871968707446785302475, −2.89594505070418331948134393795, −1.08709866809646872358259885576, 1.62402751970546023174240358176, 3.30406957724403686671233588696, 3.83575061896005219278943769166, 4.72300335513639834811327317837, 5.41360962153666374567616009016, 6.35720057702163874528058167957, 7.66147355821307755230758828316, 8.565603232406119711050396906439, 9.408013173610821173922495688457, 10.07484401905201351289175688225

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