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Properties

Label 2-378-189.5-c1-0-12
Degree $2$
Conductor $378$
Sign $0.983 - 0.181i$
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  + (−0.984 + 0.173i)2-s + (1.22 + 1.22i)3-s + (0.939 − 0.342i)4-s + (0.204 − 1.16i)5-s + (−1.41 − 0.996i)6-s + (1.70 − 2.02i)7-s + (−0.866 + 0.5i)8-s + (−0.0148 + 2.99i)9-s + 1.17i·10-s + (1.39 − 0.246i)11-s + (1.56 + 0.735i)12-s + (2.12 − 2.53i)13-s + (−1.32 + 2.28i)14-s + (1.67 − 1.16i)15-s + (0.766 − 0.642i)16-s + 0.266·17-s + ⋯
L(s)  = 1  + (−0.696 + 0.122i)2-s + (0.705 + 0.708i)3-s + (0.469 − 0.171i)4-s + (0.0916 − 0.519i)5-s + (−0.578 − 0.407i)6-s + (0.645 − 0.764i)7-s + (−0.306 + 0.176i)8-s + (−0.00494 + 0.999i)9-s + 0.373i·10-s + (0.420 − 0.0741i)11-s + (0.452 + 0.212i)12-s + (0.590 − 0.703i)13-s + (−0.355 + 0.611i)14-s + (0.432 − 0.301i)15-s + (0.191 − 0.160i)16-s + 0.0645·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.983 - 0.181i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.983 - 0.181i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37734 + 0.126178i\)
\(L(\frac12)\) \(\approx\) \(1.37734 + 0.126178i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 - 0.173i)T \)
3 \( 1 + (-1.22 - 1.22i)T \)
7 \( 1 + (-1.70 + 2.02i)T \)
good5 \( 1 + (-0.204 + 1.16i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-1.39 + 0.246i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (-2.12 + 2.53i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 - 0.266T + 17T^{2} \)
19 \( 1 + 5.14iT - 19T^{2} \)
23 \( 1 + (3.20 - 3.81i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (0.521 + 0.621i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (-3.38 - 9.28i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (-0.802 - 1.38i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.31 + 4.46i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.79 + 0.651i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-10.6 - 3.87i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (8.07 - 4.66i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.24 - 1.04i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (2.63 - 7.22i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.76 + 9.99i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (5.85 + 3.38i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (9.49 + 5.48i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.26 - 7.20i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-1.27 + 1.06i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + 3.50T + 89T^{2} \)
97 \( 1 + (3.66 - 10.0i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.92962135635419157438733619162, −10.48744278063649854579516678908, −9.387258819131401432327531297239, −8.691685406176333088936301846506, −7.948474425747845943164927596388, −6.97031299539209050168918695517, −5.41947607431551340264721909972, −4.40516792884108086911835144620, −3.12309517227222237484530760729, −1.37162154413452249973882886702, 1.60220129480560887275927892485, 2.61829960348588968466293907149, 4.02620429677491668602394552264, 5.97777898333525131740406089089, 6.70570544225938381607398389841, 7.87483708701008329551454419689, 8.481215808109627365704013980585, 9.323403013552552485275201554837, 10.29061273223538620232860897916, 11.49508666401682924967863535020

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