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Properties

Label 2-378-21.5-c1-0-9
Degree $2$
Conductor $378$
Sign $0.965 + 0.259i$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (1.22 − 2.12i)5-s + (−1 − 2.44i)7-s + 0.999i·8-s + (2.12 − 1.22i)10-s + (3.67 − 2.12i)11-s + 0.717i·13-s + (0.358 − 2.62i)14-s + (−0.5 + 0.866i)16-s + (1.22 + 2.12i)17-s + (−4.24 − 2.44i)19-s + 2.44·20-s + 4.24·22-s + (5.19 + 3i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.547 − 0.948i)5-s + (−0.377 − 0.925i)7-s + 0.353i·8-s + (0.670 − 0.387i)10-s + (1.10 − 0.639i)11-s + 0.198i·13-s + (0.0958 − 0.700i)14-s + (−0.125 + 0.216i)16-s + (0.297 + 0.514i)17-s + (−0.973 − 0.561i)19-s + 0.547·20-s + 0.904·22-s + (1.08 + 0.625i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.965 + 0.259i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.965 + 0.259i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02100 - 0.267049i\)
\(L(\frac12)\) \(\approx\) \(2.02100 - 0.267049i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (1 + 2.44i)T \)
good5 \( 1 + (-1.22 + 2.12i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.67 + 2.12i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.717iT - 13T^{2} \)
17 \( 1 + (-1.22 - 2.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.24 + 2.44i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.75iT - 29T^{2} \)
31 \( 1 + (7.86 - 4.54i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.62 - 4.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.44T + 41T^{2} \)
43 \( 1 - 7T + 43T^{2} \)
47 \( 1 + (-6.42 + 11.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (12.5 - 7.24i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.22 + 2.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.62 - 2.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.74 + 11.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.7iT - 71T^{2} \)
73 \( 1 + (4.75 - 2.74i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.378 - 0.655i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 + (-1.52 + 2.63i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.16iT - 97T^{2} \)
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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

−11.37418486890661644057442616860, −10.53960807927572189458715691915, −9.215034580542152610879074388755, −8.731702260864713522446823821123, −7.32129762957018098599260972407, −6.48692310076617168282898950078, −5.47748161190858307956904507325, −4.39119045265655951491120401214, −3.40981879657180536897900609000, −1.37216278210050611772860377394, 2.04013866467793594819661442217, 3.04042345292778083799062935256, 4.33089915590753635755573465160, 5.73975589114932227377490824639, 6.39720074459873451046111595409, 7.34174595592807526606205457549, 8.973468778612372450212796162158, 9.641028168194390643250493815706, 10.63992321482116228109233128985, 11.39737431712904482963449389910

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