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Properties

Label 2-378-63.25-c1-0-3
Degree $2$
Conductor $378$
Sign $0.997 + 0.0709i$
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 + (0.794 − 1.37i)5-s + (1.23 + 2.33i)7-s + 8-s + (0.794 − 1.37i)10-s + (−0.794 − 1.37i)11-s + (2.40 + 4.16i)13-s + (1.23 + 2.33i)14-s + 16-s + (2.69 − 4.67i)17-s + (−3.54 − 6.14i)19-s + (0.794 − 1.37i)20-s + (−0.794 − 1.37i)22-s + (0.150 − 0.260i)23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.355 − 0.615i)5-s + (0.468 + 0.883i)7-s + 0.353·8-s + (0.251 − 0.434i)10-s + (−0.239 − 0.414i)11-s + (0.667 + 1.15i)13-s + (0.331 + 0.624i)14-s + 0.250·16-s + (0.654 − 1.13i)17-s + (−0.814 − 1.41i)19-s + (0.177 − 0.307i)20-s + (−0.169 − 0.293i)22-s + (0.0313 − 0.0542i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.997 + 0.0709i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.997 + 0.0709i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23758 - 0.0794359i\)
\(L(\frac12)\) \(\approx\) \(2.23758 - 0.0794359i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 - 2.33i)T \)
good5 \( 1 + (-0.794 + 1.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.794 + 1.37i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.40 - 4.16i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.69 + 4.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.54 + 6.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.150 + 0.260i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.13 - 7.16i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.71T + 31T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.93 + 5.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.833 - 1.44i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.66T + 47T^{2} \)
53 \( 1 + (2.44 - 4.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.47T + 59T^{2} \)
61 \( 1 + 4.47T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + (-8.02 + 13.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 8.38T + 79T^{2} \)
83 \( 1 + (1.18 - 2.04i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.60 + 2.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.712 + 1.23i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.45807417523839563883789395409, −10.79224367098339540547287642455, −9.115367137360700961327604945559, −8.932372224847065297202813349857, −7.47144583719307363943543626941, −6.36877179045132751880671752955, −5.32152919242022192799525238896, −4.66724574913227276869726483909, −3.10888205063618535068314362421, −1.73428687845461665640461730786, 1.75608942409201619663298673545, 3.34584628962931518742568875030, 4.26923249519894237669821071818, 5.66501125091915023492950419213, 6.37011447390536926661186394513, 7.65033558965519131583075128018, 8.221733524285640817547949021533, 10.09671044062488280026488298683, 10.42338423939332468143614813032, 11.26959918902398902739055481855

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