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Properties

Label 2-378-7.4-c3-0-24
Degree $2$
Conductor $378$
Sign $-0.817 + 0.575i$
Analytic cond. $22.3027$
Root an. cond. $4.72257$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (−1.99 + 3.46i)4-s + (2.21 + 3.83i)5-s + (−11.5 − 14.4i)7-s + 7.99·8-s + (4.43 − 7.67i)10-s + (19.8 − 34.3i)11-s + 37.4·13-s + (−13.4 + 34.5i)14-s + (−8 − 13.8i)16-s + (−57.0 + 98.7i)17-s + (67.5 + 117. i)19-s − 17.7·20-s − 79.2·22-s + (−96.1 − 166. i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.198 + 0.343i)5-s + (−0.626 − 0.779i)7-s + 0.353·8-s + (0.140 − 0.242i)10-s + (0.543 − 0.940i)11-s + 0.799·13-s + (−0.256 + 0.659i)14-s + (−0.125 − 0.216i)16-s + (−0.813 + 1.40i)17-s + (0.815 + 1.41i)19-s − 0.198·20-s − 0.768·22-s + (−0.872 − 1.51i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.817 + 0.575i$
Analytic conductor: \(22.3027\)
Root analytic conductor: \(4.72257\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :3/2),\ -0.817 + 0.575i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9689678646\)
\(L(\frac12)\) \(\approx\) \(0.9689678646\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
3 \( 1 \)
7 \( 1 + (11.5 + 14.4i)T \)
good5 \( 1 + (-2.21 - 3.83i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-19.8 + 34.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 37.4T + 2.19e3T^{2} \)
17 \( 1 + (57.0 - 98.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-67.5 - 117. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (96.1 + 166. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 129.T + 2.43e4T^{2} \)
31 \( 1 + (-122. + 211. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (136. + 236. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 365.T + 6.89e4T^{2} \)
43 \( 1 - 57.2T + 7.95e4T^{2} \)
47 \( 1 + (172. + 298. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-300. + 521. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (15.7 - 27.3i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (431. + 746. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-341. + 591. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 664.T + 3.57e5T^{2} \)
73 \( 1 + (471. - 816. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-1.66 - 2.87i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 54.0T + 5.71e5T^{2} \)
89 \( 1 + (142. + 247. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 240.T + 9.12e5T^{2} \)
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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

−10.47840772530459866352435670099, −9.965258085927080716454999396269, −8.705156373152597608885200558244, −8.059912138582715211173009681937, −6.62181456892142569127407246232, −5.99070303818976115903365580098, −4.05423714799968224296852002560, −3.47453950028683912279675277911, −1.86029017400463803002205531444, −0.38807532547051525168841310866, 1.43011263435951765636021842113, 3.06070722162938749261009602859, 4.69355014377950942455069666506, 5.56294647098442290867339609625, 6.71194716747147579099828147441, 7.36727157828119950413232438156, 8.896720536398203394518143825611, 9.176619083208163913536845703325, 10.03459567965832071253601351017, 11.43638097052935814408946288672

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