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Properties

Label 2-378-63.58-c1-0-6
Degree $2$
Conductor $378$
Sign $0.143 + 0.989i$
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  + 2-s + 4-s + (−1.59 − 2.75i)5-s + (−2.56 − 0.658i)7-s + 8-s + (−1.59 − 2.75i)10-s + (1.59 − 2.75i)11-s + (2.85 − 4.93i)13-s + (−2.56 − 0.658i)14-s + 16-s + (0.760 + 1.31i)17-s + (−0.641 + 1.11i)19-s + (−1.59 − 2.75i)20-s + (1.59 − 2.75i)22-s + (1.11 + 1.93i)23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.711 − 1.23i)5-s + (−0.968 − 0.249i)7-s + 0.353·8-s + (−0.503 − 0.871i)10-s + (0.479 − 0.830i)11-s + (0.790 − 1.36i)13-s + (−0.684 − 0.176i)14-s + 0.250·16-s + (0.184 + 0.319i)17-s + (−0.147 + 0.254i)19-s + (−0.355 − 0.616i)20-s + (0.339 − 0.587i)22-s + (0.233 + 0.404i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.143 + 0.989i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.143 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24203 - 1.07492i\)
\(L(\frac12)\) \(\approx\) \(1.24203 - 1.07492i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + (2.56 + 0.658i)T \)
good5 \( 1 + (1.59 + 2.75i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.59 + 2.75i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.85 + 4.93i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.760 - 1.31i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.641 - 1.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.11 - 1.93i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.54 - 6.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.42T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.80 + 4.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.41 - 5.91i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.82T + 47T^{2} \)
53 \( 1 + (1.02 + 1.78i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.12T + 59T^{2} \)
61 \( 1 - 3.12T + 61T^{2} \)
67 \( 1 - 10.9T + 67T^{2} \)
71 \( 1 + 8.69T + 71T^{2} \)
73 \( 1 + (2.48 + 4.30i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 4.13T + 79T^{2} \)
83 \( 1 + (-4.03 - 6.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.112 - 0.195i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.42 - 12.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.25240490562333467277108571867, −10.47977003940268642389998213666, −9.147420009167003228359417107896, −8.393855637531798404792065034886, −7.37681615111493128729124342001, −6.07878265916212801829735997984, −5.32562995097282969507034760154, −3.94746548860829451773895114774, −3.29951961768226093161864040708, −0.933281394689002347831166355662, 2.36429474107570020497919387375, 3.55586632460255318295741375386, 4.33193110203303504909978094090, 6.02501738194667865983584268696, 6.80677454649809242206371047735, 7.35046269869214208483402399477, 8.898661251143797197893127311219, 9.871740531715788824950773237143, 10.92097246224469010923652025123, 11.61671305568902170277914884266

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