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Properties

Label 2-378-9.7-c1-0-4
Degree $2$
Conductor $378$
Sign $-0.569 + 0.821i$
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 + (−0.724 − 1.25i)5-s + (0.5 − 0.866i)7-s − 0.999·8-s − 1.44·10-s + (1 − 1.73i)11-s + (−2.44 − 4.24i)13-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 2·17-s + 2.55·19-s + (−0.724 + 1.25i)20-s + (−0.999 − 1.73i)22-s + (−0.5 − 0.866i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.324 − 0.561i)5-s + (0.188 − 0.327i)7-s − 0.353·8-s − 0.458·10-s + (0.301 − 0.522i)11-s + (−0.679 − 1.17i)13-s + (−0.133 − 0.231i)14-s + (−0.125 + 0.216i)16-s − 0.485·17-s + 0.585·19-s + (−0.162 + 0.280i)20-s + (−0.213 − 0.369i)22-s + (−0.104 − 0.180i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.569 + 0.821i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.569 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.631701 - 1.20668i\)
\(L(\frac12)\) \(\approx\) \(0.631701 - 1.20668i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.724 + 1.25i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.44 + 4.24i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 2.55T + 19T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.44 - 5.97i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 11.7T + 37T^{2} \)
41 \( 1 + (-4.89 - 8.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.44 - 5.97i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.89 + 8.48i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.27 - 5.67i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.44 - 11.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.101T + 71T^{2} \)
73 \( 1 + 6.89T + 73T^{2} \)
79 \( 1 + (-0.949 + 1.64i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1 + 1.73i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 16.8T + 89T^{2} \)
97 \( 1 + (1.44 - 2.51i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.18272822593930734220308941788, −10.26641585509377529613766846008, −9.336119697365867413454268928704, −8.329511368106140275767813296213, −7.39864112329380115867568751883, −5.96830505884630385383010518726, −4.96646380610646230688410994537, −3.97174183351867101845986745049, −2.69954442091632307699708298370, −0.850324719538603196320751462879, 2.30096668731260982282408203411, 3.81865948812541205442946576738, 4.79790878974330729867877771018, 6.00097999082114029471775925221, 7.07150880346680772691937068749, 7.59217796900276123980170463359, 8.956877084560130948535618955995, 9.606490798697968293005371550516, 10.97516917689667702585054392558, 11.75841635180291790749992382749

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