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Properties

Label 2-378-3.2-c4-0-22
Degree $2$
Conductor $378$
Sign $i$
Analytic cond. $39.0738$
Root an. cond. $6.25090$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82i·2-s − 8.00·4-s − 9.19i·5-s + 18.5·7-s + 22.6i·8-s − 25.9·10-s − 68.1i·11-s + 212.·13-s − 52.3i·14-s + 64.0·16-s + 340. i·17-s − 176.·19-s + 73.5i·20-s − 192.·22-s + 241. i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.500·4-s − 0.367i·5-s + 0.377·7-s + 0.353i·8-s − 0.259·10-s − 0.563i·11-s + 1.25·13-s − 0.267i·14-s + 0.250·16-s + 1.17i·17-s − 0.488·19-s + 0.183i·20-s − 0.398·22-s + 0.456i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $i$
Analytic conductor: \(39.0738\)
Root analytic conductor: \(6.25090\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :2),\ i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.043548408\)
\(L(\frac12)\) \(\approx\) \(2.043548408\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82iT \)
3 \( 1 \)
7 \( 1 - 18.5T \)
good5 \( 1 + 9.19iT - 625T^{2} \)
11 \( 1 + 68.1iT - 1.46e4T^{2} \)
13 \( 1 - 212.T + 2.85e4T^{2} \)
17 \( 1 - 340. iT - 8.35e4T^{2} \)
19 \( 1 + 176.T + 1.30e5T^{2} \)
23 \( 1 - 241. iT - 2.79e5T^{2} \)
29 \( 1 + 549. iT - 7.07e5T^{2} \)
31 \( 1 - 980.T + 9.23e5T^{2} \)
37 \( 1 - 743.T + 1.87e6T^{2} \)
41 \( 1 + 1.96e3iT - 2.82e6T^{2} \)
43 \( 1 + 1.35e3T + 3.41e6T^{2} \)
47 \( 1 - 1.32e3iT - 4.87e6T^{2} \)
53 \( 1 + 4.06e3iT - 7.89e6T^{2} \)
59 \( 1 + 3.52e3iT - 1.21e7T^{2} \)
61 \( 1 - 5.20e3T + 1.38e7T^{2} \)
67 \( 1 - 2.24e3T + 2.01e7T^{2} \)
71 \( 1 + 7.43e3iT - 2.54e7T^{2} \)
73 \( 1 + 1.05e3T + 2.83e7T^{2} \)
79 \( 1 - 3.14e3T + 3.89e7T^{2} \)
83 \( 1 - 467. iT - 4.74e7T^{2} \)
89 \( 1 - 2.86e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.58e4T + 8.85e7T^{2} \)
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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

−10.74348452933952964352425451807, −9.688731448687857982890043786821, −8.508046866679911456528868949458, −8.232966662389218507201474020770, −6.53102306295946533827240259043, −5.54647836056254814071753726388, −4.32545283920799552824860322883, −3.38532514986569486799784649051, −1.86768549578199734363258936981, −0.74713715857206539878913860499, 1.05628378193242443144103720114, 2.78203060534989175820819682109, 4.19690710491412862262798159872, 5.16134815013939734633507293265, 6.37200906478356498865824436107, 7.07233914911651824723803718544, 8.166441024132307322244057959708, 8.918431726683895354778378058109, 9.997816347758387804356073616027, 10.91706765278967063778854725868

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