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Properties

Label 2-378-7.4-c3-0-14
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

\(L(2)\) \(\approx\) \(2.015943894\)
\(L(\frac12)\) \(\approx\) \(2.015943894\)
\(L(\frac{5}{2})\) 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

−11.06460660947336243017449644148, −9.891649158680089619190705488482, −9.301554701717896432820323999264, −7.75709819149894789307150302894, −7.48320002647072965813163284853, −6.21263009948866724724087320831, −5.18799367542909343180292363349, −4.12236570069737637796489409379, −3.04307590418750820041490286876, −0.912976215873091690015010419648, 0.923940849618747846577788330444, 2.79511338873265944527880194296, 3.36768892054348674353313745940, 4.92530928836717821252721820681, 5.94121865674349639011931225890, 6.80855576474852262258706163834, 8.404526170838382511948945282684, 8.928659233772794812844507769164, 10.25774982279761014755379286541, 10.84157363367038197587402271445

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