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Properties

Label 2-2368-148.47-c0-0-4
Degree $2$
Conductor $2368$
Sign $0.929 + 0.367i$
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  + (0.866 − 0.5i)3-s + (0.5 + 0.866i)5-s + (0.866 − 0.5i)7-s − 2i·11-s + (0.5 + 0.866i)13-s + (0.866 + 0.499i)15-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.499 − 0.866i)21-s + i·27-s − 2·29-s + (−1 − 1.73i)33-s + (0.866 + 0.499i)35-s − 37-s + (0.866 + 0.499i)39-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (0.5 + 0.866i)5-s + (0.866 − 0.5i)7-s − 2i·11-s + (0.5 + 0.866i)13-s + (0.866 + 0.499i)15-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.499 − 0.866i)21-s + i·27-s − 2·29-s + (−1 − 1.73i)33-s + (0.866 + 0.499i)35-s − 37-s + (0.866 + 0.499i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $0.929 + 0.367i$
Analytic conductor: \(1.18178\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2368,\ (\ :0),\ 0.929 + 0.367i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.869368856\)
\(L(\frac12)\) \(\approx\) \(1.869368856\)
\(L(1)\) 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 \( 1 + T \)
good3 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + 2iT - T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 2T + T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.791727077733909738768737060548, −8.441695560513222399572565971968, −7.61791807301268562254960636235, −6.95759386499169477674535454774, −6.05743272109447158417294101558, −5.36683575030455239699697472845, −4.00709482799641235305153596498, −3.24499449547963333122896442741, −2.36335089396187761699306087235, −1.40337443738563565521003348559, 1.66440059975471049793532704093, 2.26014002149278191934082720767, 3.61902627966745865246345068292, 4.38802514804946538166965968105, 5.20149898173481744307147545045, 5.82612196618977955278687389276, 7.09245786586405531910866837839, 7.916332785684672492835584950936, 8.615552697203583398792778413247, 9.089541721298149218903596198713

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