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Properties

Label 2-378-7.4-c1-0-1
Degree $2$
Conductor $378$
Sign $-0.605 - 0.795i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−2 + 1.73i)7-s − 0.999·8-s + (−3 + 5.19i)11-s + 5·13-s + (−2.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (−3 + 5.19i)17-s + (2 + 3.46i)19-s − 6·22-s + (−3 − 5.19i)23-s + (2.5 − 4.33i)25-s + (2.5 + 4.33i)26-s + (−0.5 − 2.59i)28-s + 6·29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.755 + 0.654i)7-s − 0.353·8-s + (−0.904 + 1.56i)11-s + 1.38·13-s + (−0.668 − 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.727 + 1.26i)17-s + (0.458 + 0.794i)19-s − 1.27·22-s + (−0.625 − 1.08i)23-s + (0.5 − 0.866i)25-s + (0.490 + 0.849i)26-s + (−0.0944 − 0.490i)28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.565873 + 1.14150i\)
\(L(\frac12)\) \(\approx\) \(0.565873 + 1.14150i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (2 - 1.73i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 17T + 97T^{2} \)
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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

−12.05602328436749953487507509593, −10.57837394671706512395657219525, −9.918004128200334898047092817731, −8.645058975598650243652514653308, −8.061332521392267183355566815387, −6.65944530890540019030159135597, −6.13183919178818791994689393089, −4.87961467951190439939380869548, −3.78199205398717790541895117723, −2.30769262666340611120845357023, 0.77667562379779309584132223030, 2.91172864827382921688936916286, 3.64707567206504421166713519353, 5.09452636472162406048197616185, 6.08706715299460033565153939227, 7.14638130280981763896625698944, 8.470900719350957092343565843175, 9.285271695669495452058939774900, 10.38805947456277925055209132771, 11.08203437333440760215267638765

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