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Properties

Label 2-2368-37.17-c0-0-0
Degree $2$
Conductor $2368$
Sign $-0.0237 - 0.999i$
Analytic cond. $1.18178$
Root an. cond. $1.08709$
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  + (0.142 + 1.63i)5-s + (−0.939 − 0.342i)9-s + (1.75 + 0.816i)13-s + (0.0736 + 0.157i)17-s + (−1.65 + 0.292i)25-s + (−0.816 − 0.218i)29-s + (0.342 + 0.939i)37-s + (0.524 + 1.43i)41-s + (0.424 − 1.58i)45-s + (−0.173 − 0.984i)49-s + (−1.32 + 1.11i)53-s + (−0.766 + 1.64i)61-s + (−1.08 + 2.97i)65-s − i·73-s + (0.766 + 0.642i)81-s + ⋯
L(s)  = 1  + (0.142 + 1.63i)5-s + (−0.939 − 0.342i)9-s + (1.75 + 0.816i)13-s + (0.0736 + 0.157i)17-s + (−1.65 + 0.292i)25-s + (−0.816 − 0.218i)29-s + (0.342 + 0.939i)37-s + (0.524 + 1.43i)41-s + (0.424 − 1.58i)45-s + (−0.173 − 0.984i)49-s + (−1.32 + 1.11i)53-s + (−0.766 + 1.64i)61-s + (−1.08 + 2.97i)65-s − i·73-s + (0.766 + 0.642i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0237 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0237 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $-0.0237 - 0.999i$
Analytic conductor: \(1.18178\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (1793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2368,\ (\ :0),\ -0.0237 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.132672607\)
\(L(\frac12)\) \(\approx\) \(1.132672607\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (-0.342 - 0.939i)T \)
good3 \( 1 + (0.939 + 0.342i)T^{2} \)
5 \( 1 + (-0.142 - 1.63i)T + (-0.984 + 0.173i)T^{2} \)
7 \( 1 + (0.173 + 0.984i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-1.75 - 0.816i)T + (0.642 + 0.766i)T^{2} \)
17 \( 1 + (-0.0736 - 0.157i)T + (-0.642 + 0.766i)T^{2} \)
19 \( 1 + (0.342 - 0.939i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.816 + 0.218i)T + (0.866 + 0.5i)T^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 + (-0.524 - 1.43i)T + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (1.32 - 1.11i)T + (0.173 - 0.984i)T^{2} \)
59 \( 1 + (0.984 + 0.173i)T^{2} \)
61 \( 1 + (0.766 - 1.64i)T + (-0.642 - 0.766i)T^{2} \)
67 \( 1 + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (-0.939 - 0.342i)T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 + (-0.984 + 0.173i)T^{2} \)
83 \( 1 + (0.766 + 0.642i)T^{2} \)
89 \( 1 + (-0.173 + 1.98i)T + (-0.984 - 0.173i)T^{2} \)
97 \( 1 + (-1.75 + 0.469i)T + (0.866 - 0.5i)T^{2} \)
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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

−9.337594484474295683235223987968, −8.608604289092814368869222453222, −7.79436087751693092087540514705, −6.90575272034489007779893859858, −6.10444395230467232854845777787, −5.96820552667795383434246602447, −4.39590503361247192684467944615, −3.41638985942806182652361679867, −2.91325130672881974487510345831, −1.67382314886382594167835765077, 0.796957254173391798857415290309, 1.93760895102200272764976407752, 3.30721540438529872019440115631, 4.15327978145002244599892271337, 5.25983954880819794668096294515, 5.59102191366050918339771787016, 6.41998261824713632365039117558, 7.902324106316434319359416443864, 8.135326520358907180846239646064, 9.104343716696493956562342740767

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