Properties

Label 2-378-27.25-c1-0-11
Degree $2$
Conductor $378$
Sign $-0.771 + 0.636i$
Analytic cond. $3.01834$
Root an. cond. $1.73733$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 − 0.984i)2-s + (−1.73 + 0.0288i)3-s + (−0.939 − 0.342i)4-s + (1.44 − 1.21i)5-s + (−0.272 + 1.71i)6-s + (0.939 − 0.342i)7-s + (−0.5 + 0.866i)8-s + (2.99 − 0.100i)9-s + (−0.941 − 1.63i)10-s + (−2.52 − 2.11i)11-s + (1.63 + 0.565i)12-s + (−0.345 − 1.96i)13-s + (−0.173 − 0.984i)14-s + (−2.46 + 2.13i)15-s + (0.766 + 0.642i)16-s + (−1.72 − 2.98i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (−0.999 + 0.0166i)3-s + (−0.469 − 0.171i)4-s + (0.645 − 0.541i)5-s + (−0.111 + 0.698i)6-s + (0.355 − 0.129i)7-s + (−0.176 + 0.306i)8-s + (0.999 − 0.0333i)9-s + (−0.297 − 0.515i)10-s + (−0.761 − 0.638i)11-s + (0.472 + 0.163i)12-s + (−0.0959 − 0.544i)13-s + (−0.0464 − 0.263i)14-s + (−0.636 + 0.552i)15-s + (0.191 + 0.160i)16-s + (−0.417 − 0.723i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.771 + 0.636i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.771 + 0.636i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.315525 - 0.878010i\)
\(L(\frac12)\) \(\approx\) \(0.315525 - 0.878010i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + (1.73 - 0.0288i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
good5 \( 1 + (-1.44 + 1.21i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (2.52 + 2.11i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.345 + 1.96i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.72 + 2.98i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.13 + 1.96i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.20 + 1.16i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.895 + 5.07i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (8.22 + 2.99i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-2.08 - 3.61i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.18 + 6.69i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-4.71 - 3.96i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-0.423 + 0.154i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 - 13.8T + 53T^{2} \)
59 \( 1 + (-6.94 + 5.82i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-12.7 + 4.65i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.997 - 5.65i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-2.17 - 3.76i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.36 - 12.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.07 - 11.7i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-0.508 + 2.88i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-0.135 + 0.235i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.24 + 1.04i)T + (16.8 + 95.5i)T^{2} \)
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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

−11.15772620763585644985281687428, −10.22482704497083704344061907734, −9.521894689415002847499407863985, −8.359858913779399171180677089833, −7.15394862320821816948537171218, −5.67010700006293185261973954175, −5.28763335589803899371019333857, −4.07422077800139305258417017486, −2.31675591900469192682570380900, −0.68394596072725699033237983099, 1.99673902591658931367177098777, 4.04621041165899798750905703485, 5.19213788268252627856475513402, 5.93014518969294105752054122013, 6.87817581823423885013883979088, 7.64895636697090325168612858958, 8.967578709453441303387709727965, 10.11256828898090260816621154048, 10.62735155153625364875546137135, 11.76993355142354154741504851572

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