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Properties

Label 2-378-21.20-c1-0-6
Degree $2$
Conductor $378$
Sign $0.654 - 0.755i$
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  + i·2-s − 4-s + 3.46·5-s + (2 + 1.73i)7-s − i·8-s + 3.46i·10-s − 6i·11-s − 1.73i·13-s + (−1.73 + 2i)14-s + 16-s − 1.73·17-s + 6.92i·19-s − 3.46·20-s + 6·22-s + 3i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 1.54·5-s + (0.755 + 0.654i)7-s − 0.353i·8-s + 1.09i·10-s − 1.80i·11-s − 0.480i·13-s + (−0.462 + 0.534i)14-s + 0.250·16-s − 0.420·17-s + 1.58i·19-s − 0.774·20-s + 1.27·22-s + 0.625i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.654 - 0.755i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.654 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56819 + 0.716430i\)
\(L(\frac12)\) \(\approx\) \(1.56819 + 0.716430i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (-2 - 1.73i)T \)
good5 \( 1 - 3.46T + 5T^{2} \)
11 \( 1 + 6iT - 11T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 + 1.73T + 17T^{2} \)
19 \( 1 - 6.92iT - 19T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 - 5.19iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 + 11T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 + 8.66T + 59T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 + 7T + 67T^{2} \)
71 \( 1 + 3iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 - 5.19T + 89T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−11.38534070428826074083399528369, −10.44171597416991846699776687910, −9.511582975420566277813295825902, −8.599984572855352135130624756224, −7.976214258126713115784085842596, −6.33805318076580756535377255225, −5.80658807229212780804422961734, −5.09326916409939077370862112546, −3.26702218997399691608563766513, −1.65773064102172855830350171016, 1.61915126982252147919498589474, 2.45996292262262875043062996812, 4.44823347395466669575215431044, 5.00730869776337642163543849441, 6.50200967289606503170204745064, 7.39251321746416321140727485722, 8.871128503042850255390060382313, 9.586658767515454854847101774587, 10.29877688104426334052135431195, 11.08121340627381757629314675751

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