Properties

Label 2-378-27.16-c1-0-11
Degree $2$
Conductor $378$
Sign $0.723 - 0.690i$
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  + (0.766 + 0.642i)2-s + (1.73 − 0.0705i)3-s + (0.173 + 0.984i)4-s + (−0.918 − 0.334i)5-s + (1.37 + 1.05i)6-s + (−0.173 + 0.984i)7-s + (−0.500 + 0.866i)8-s + (2.99 − 0.244i)9-s + (−0.488 − 0.846i)10-s + (3.68 − 1.34i)11-s + (0.369 + 1.69i)12-s + (0.336 − 0.282i)13-s + (−0.766 + 0.642i)14-s + (−1.61 − 0.513i)15-s + (−0.939 + 0.342i)16-s + (1.75 + 3.04i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (0.999 − 0.0407i)3-s + (0.0868 + 0.492i)4-s + (−0.410 − 0.149i)5-s + (0.559 + 0.432i)6-s + (−0.0656 + 0.372i)7-s + (−0.176 + 0.306i)8-s + (0.996 − 0.0813i)9-s + (−0.154 − 0.267i)10-s + (1.11 − 0.404i)11-s + (0.106 + 0.488i)12-s + (0.0933 − 0.0783i)13-s + (−0.204 + 0.171i)14-s + (−0.416 − 0.132i)15-s + (−0.234 + 0.0855i)16-s + (0.425 + 0.737i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.723 - 0.690i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.723 - 0.690i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19443 + 0.879196i\)
\(L(\frac12)\) \(\approx\) \(2.19443 + 0.879196i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (-1.73 + 0.0705i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
good5 \( 1 + (0.918 + 0.334i)T + (3.83 + 3.21i)T^{2} \)
11 \( 1 + (-3.68 + 1.34i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-0.336 + 0.282i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-1.75 - 3.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.72 - 6.45i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.806 + 4.57i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (3.40 + 2.85i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.888 + 5.04i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (4.10 + 7.11i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.827 + 0.694i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (10.9 - 3.98i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-0.138 + 0.784i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 8.30T + 53T^{2} \)
59 \( 1 + (-0.398 - 0.145i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.275 + 1.56i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-5.61 + 4.71i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (5.65 + 9.79i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.18 - 12.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.48 - 7.12i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-6.15 - 5.16i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-1.02 + 1.77i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.7 - 4.99i)T + (74.3 - 62.3i)T^{2} \)
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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

−11.83723286803183525645933020473, −10.48453755323016398624698771024, −9.413895855634658979689883519489, −8.386783969650679796166023974791, −7.988159996027664123619988950290, −6.67594564967216857661209168401, −5.81532277990660959576675473289, −4.13328748138840933885555006549, −3.66375313102556280623245753925, −2.02643026437266168033626262021, 1.64743769086139097390075987842, 3.15614605164498392390201810201, 3.96364843387482496618162008053, 5.00695715921494533439486002361, 6.75295862537992297181002109442, 7.31831575269044909156208142498, 8.680942730414472592543646403511, 9.433138409042048131534008602815, 10.31379318762311989807880079802, 11.41816051997051784731245691065

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