Properties

Label 2-378-7.2-c1-0-6
Degree $2$
Conductor $378$
Sign $0.0633 + 0.997i$
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 + (0.822 − 1.42i)5-s + 2.64·7-s − 0.999·8-s + (−0.822 − 1.42i)10-s + (−0.822 − 1.42i)11-s + 0.645·13-s + (1.32 − 2.29i)14-s + (−0.5 + 0.866i)16-s + (−0.822 − 1.42i)17-s + (−1 + 1.73i)19-s − 1.64·20-s − 1.64·22-s + (4.64 − 8.04i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.368 − 0.637i)5-s + 0.999·7-s − 0.353·8-s + (−0.260 − 0.450i)10-s + (−0.248 − 0.429i)11-s + 0.179·13-s + (0.353 − 0.612i)14-s + (−0.125 + 0.216i)16-s + (−0.199 − 0.345i)17-s + (−0.229 + 0.397i)19-s − 0.368·20-s − 0.350·22-s + (0.968 − 1.67i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.0633 + 0.997i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.0633 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28803 - 1.20888i\)
\(L(\frac12)\) \(\approx\) \(1.28803 - 1.20888i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.64T \)
good5 \( 1 + (-0.822 + 1.42i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.822 + 1.42i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.645T + 13T^{2} \)
17 \( 1 + (0.822 + 1.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.64 + 8.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.64T + 29T^{2} \)
31 \( 1 + (0.322 + 0.559i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.96 - 3.40i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.93T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (5.46 - 9.47i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.82 - 11.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.32 - 10.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.14 + 7.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + (-5.29 - 9.16i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.61 + 13.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.70T + 83T^{2} \)
89 \( 1 + (5.46 - 9.47i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.58T + 97T^{2} \)
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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

−11.07173886204494887552850004518, −10.55941451273899152886505052272, −9.220978817871921792699223432308, −8.638184703580454505720409103678, −7.48508814832582750000513865064, −6.01484335420599004594830298439, −5.08921653723047640987059495213, −4.23653870630519491705674900881, −2.64666046806064661209124295917, −1.24359319753871805406851490265, 2.05215780553305001286438904907, 3.60218702640919767909995378601, 4.88897227986472045382410632931, 5.73625978484083783714330917461, 6.93515949454679271379175924539, 7.62745298964422116029932758250, 8.683580226059335918136615827409, 9.672585138842230038630824805344, 10.86953950159731328486419538211, 11.42496109035063469695100622287

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