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Properties

Label 2-2368-37.31-c0-0-3
Degree $2$
Conductor $2368$
Sign $0.763 + 0.646i$
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  + (1 − i)5-s + 9-s + (1 − i)13-s + (−1 + i)17-s − i·25-s + (−1 − i)29-s + i·37-s + 2i·41-s + (1 − i)45-s − 49-s + (−1 − i)61-s − 2i·65-s − 2i·73-s + 81-s + 2i·85-s + ⋯
L(s)  = 1  + (1 − i)5-s + 9-s + (1 − i)13-s + (−1 + i)17-s − i·25-s + (−1 − i)29-s + i·37-s + 2i·41-s + (1 − i)45-s − 49-s + (−1 − i)61-s − 2i·65-s − 2i·73-s + 81-s + 2i·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $0.763 + 0.646i$
Analytic conductor: \(1.18178\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (2177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2368,\ (\ :0),\ 0.763 + 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.546237798\)
\(L(\frac12)\) \(\approx\) \(1.546237798\)
\(L(1)\) 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 \( 1 - iT \)
good3 \( 1 - T^{2} \)
5 \( 1 + (-1 + i)T - iT^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-1 + i)T - iT^{2} \)
17 \( 1 + (1 - i)T - iT^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (1 + i)T + iT^{2} \)
31 \( 1 + iT^{2} \)
41 \( 1 - 2iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (1 + i)T + iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 2iT - T^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1 - i)T + iT^{2} \)
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

−9.196562097778241961554494417584, −8.284929239869553144901895997006, −7.79508743174382955440818872755, −6.40697487213983947911926503367, −6.12268705247531482903512327788, −5.06977503320904645705622171329, −4.41258870652461727411788871985, −3.40002976583961813580292701268, −1.98018814529738583321346873258, −1.23706287393251435694974801007, 1.62551202531550292762246902933, 2.36371847893255118278924014205, 3.54209722905319536299047579880, 4.37838322348665543379014844050, 5.44433536322227225808975833846, 6.27917468871920546780846589284, 6.98523776237877350737607345706, 7.32325913756329044691443411078, 8.765388207096729514474166032960, 9.260011170254545956038431180785

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