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Properties

Label 2-378-7.4-c3-0-3
Degree $2$
Conductor $378$
Sign $-0.905 - 0.423i$
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 + (−8.05 − 13.9i)5-s + (6.77 − 17.2i)7-s − 7.99·8-s + (16.1 − 27.9i)10-s + (−6.04 + 10.4i)11-s − 39.7·13-s + (36.6 − 5.50i)14-s + (−8 − 13.8i)16-s + (−62.7 + 108. i)17-s + (62.5 + 108. i)19-s + 64.4·20-s − 24.1·22-s + (24.1 + 41.7i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.720 − 1.24i)5-s + (0.365 − 0.930i)7-s − 0.353·8-s + (0.509 − 0.882i)10-s + (−0.165 + 0.286i)11-s − 0.848·13-s + (0.699 − 0.105i)14-s + (−0.125 − 0.216i)16-s + (−0.895 + 1.55i)17-s + (0.754 + 1.30i)19-s + 0.720·20-s − 0.234·22-s + (0.218 + 0.378i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.905 - 0.423i$
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: \((2,\ 378,\ (\ :3/2),\ -0.905 - 0.423i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5628370756\)
\(L(\frac12)\) \(\approx\) \(0.5628370756\)
\(L(\frac{5}{2})\) 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 \( 1 + (-6.77 + 17.2i)T \)
good5 \( 1 + (8.05 + 13.9i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (6.04 - 10.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 39.7T + 2.19e3T^{2} \)
17 \( 1 + (62.7 - 108. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-62.5 - 108. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-24.1 - 41.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 93.0T + 2.43e4T^{2} \)
31 \( 1 + (11.3 - 19.7i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-200. - 347. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 354.T + 6.89e4T^{2} \)
43 \( 1 - 225.T + 7.95e4T^{2} \)
47 \( 1 + (124. + 214. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (167. - 290. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-91.4 + 158. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (304. + 528. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (59.1 - 102. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 81.4T + 3.57e5T^{2} \)
73 \( 1 + (-489. + 847. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-346. - 600. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.40e3T + 5.71e5T^{2} \)
89 \( 1 + (-605. - 1.04e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.33e3T + 9.12e5T^{2} \)
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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

−11.56089104195596290115963406434, −10.37834586798648999551315182372, −9.341512260549881696866469740096, −8.115929369578170467926938298869, −7.85766478417752051565335733311, −6.64936642632660279556248604919, −5.29670627687640680100309177765, −4.48526243307980339055994426412, −3.69685241576617649493626795087, −1.46684967396321380591933634379, 0.17104929620653704466590756004, 2.46492186549737317835056369111, 2.98005502645857070240003176760, 4.49232786439808818563600522418, 5.45678428960110011407503159330, 6.83368771149526294473355027044, 7.54546063867858448397759058266, 8.919605121084538204998937151656, 9.685868274190217094887991016519, 11.03159052396319865071158912973

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