Properties

Label 2-378-189.41-c1-0-4
Degree $2$
Conductor $378$
Sign $0.536 - 0.844i$
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 + (−1.41 + 0.992i)3-s + (0.939 − 0.342i)4-s + (3.28 + 2.75i)5-s + (−1.22 + 1.22i)6-s + (−2.27 − 1.34i)7-s + (0.866 − 0.5i)8-s + (1.02 − 2.81i)9-s + (3.70 + 2.14i)10-s + (2.71 + 3.22i)11-s + (−0.994 + 1.41i)12-s + (−0.922 − 0.162i)13-s + (−2.47 − 0.934i)14-s + (−7.38 − 0.649i)15-s + (0.766 − 0.642i)16-s + (−3.26 + 5.66i)17-s + ⋯
L(s)  = 1  + (0.696 − 0.122i)2-s + (−0.819 + 0.573i)3-s + (0.469 − 0.171i)4-s + (1.46 + 1.23i)5-s + (−0.500 + 0.499i)6-s + (−0.860 − 0.510i)7-s + (0.306 − 0.176i)8-s + (0.342 − 0.939i)9-s + (1.17 + 0.677i)10-s + (0.817 + 0.973i)11-s + (−0.286 + 0.409i)12-s + (−0.255 − 0.0451i)13-s + (−0.661 − 0.249i)14-s + (−1.90 − 0.167i)15-s + (0.191 − 0.160i)16-s + (−0.792 + 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.536 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.536 - 0.844i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.536 - 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62318 + 0.892125i\)
\(L(\frac12)\) \(\approx\) \(1.62318 + 0.892125i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - 0.992i)T \)
7 \( 1 + (2.27 + 1.34i)T \)
good5 \( 1 + (-3.28 - 2.75i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (-2.71 - 3.22i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.922 + 0.162i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (3.26 - 5.66i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.47 + 0.849i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.996 + 2.73i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-7.02 + 1.23i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.84 + 7.81i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (-1.18 + 2.04i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.947 + 5.37i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (3.40 - 2.85i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (0.161 + 0.0587i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 8.21iT - 53T^{2} \)
59 \( 1 + (6.04 + 5.07i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-2.33 + 6.40i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.519 + 2.94i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (5.28 + 3.05i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.852 - 0.492i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.885 + 5.02i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.395 + 2.24i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (1.22 + 2.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.53 + 1.82i)T + (-16.8 + 95.5i)T^{2} \)
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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

−11.34364820376406075747218195377, −10.51321293662249803780716787853, −10.00034286123501685454952806007, −9.344257548547079059751551988765, −7.08514787626104666954973102223, −6.42681191499328033157715783926, −5.97194680150968321202686427898, −4.56023459256623839557570295596, −3.48991689552496109306702984207, −2.06098149112520796729402950955, 1.24827092785550783585073875385, 2.73194621690915009950907267622, 4.69755445385363147046944310581, 5.50148592963017273079680994205, 6.18161265289046742094330647385, 6.92693158355358006059638076859, 8.600356040004089344239095077841, 9.354197463383194217425948897100, 10.32663435468238738397580972485, 11.65940588734386051717633327442

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