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Properties

Label 2-912-19.17-c1-0-1
Degree $2$
Conductor $912$
Sign $-0.565 - 0.824i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
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)3-s + (−0.826 + 0.300i)5-s + (−1.43 − 2.49i)7-s + (0.173 + 0.984i)9-s + (−0.918 + 1.59i)11-s + (−2.11 + 1.77i)13-s + (−0.826 − 0.300i)15-s + (−1.23 + 6.99i)17-s + (−3.93 + 1.86i)19-s + (0.500 − 2.83i)21-s + (6.19 + 2.25i)23-s + (−3.23 + 2.71i)25-s + (−0.500 + 0.866i)27-s + (−0.543 − 3.08i)29-s + (3.82 + 6.62i)31-s + ⋯
L(s)  = 1  + (0.442 + 0.371i)3-s + (−0.369 + 0.134i)5-s + (−0.544 − 0.942i)7-s + (0.0578 + 0.328i)9-s + (−0.277 + 0.479i)11-s + (−0.586 + 0.491i)13-s + (−0.213 − 0.0776i)15-s + (−0.299 + 1.69i)17-s + (−0.903 + 0.427i)19-s + (0.109 − 0.618i)21-s + (1.29 + 0.470i)23-s + (−0.647 + 0.543i)25-s + (−0.0962 + 0.166i)27-s + (−0.100 − 0.572i)29-s + (0.687 + 1.19i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.565 - 0.824i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.565 - 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.444136 + 0.842889i\)
\(L(\frac12)\) \(\approx\) \(0.444136 + 0.842889i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.766 - 0.642i)T \)
19 \( 1 + (3.93 - 1.86i)T \)
good5 \( 1 + (0.826 - 0.300i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (1.43 + 2.49i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.918 - 1.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.11 - 1.77i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (1.23 - 6.99i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-6.19 - 2.25i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.543 + 3.08i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-3.82 - 6.62i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.83T + 37T^{2} \)
41 \( 1 + (-3.05 - 2.56i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (10.7 - 3.91i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.383 + 2.17i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (2.53 + 0.924i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-1.46 + 8.28i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (0.578 + 0.210i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.638 - 3.61i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (7.00 - 2.54i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (7.66 + 6.43i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (1.23 + 1.03i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-0.492 - 0.853i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-13.0 + 10.9i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (1.02 - 5.81i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.28011395751924841430208580427, −9.714794119538679664420966263364, −8.678771561631513933045469041803, −7.87663443580767354524629541739, −7.04820787983127398248427444340, −6.25627320393458016969745695820, −4.82591730706654812319543906288, −4.04384730604352239424660490783, −3.21970403988830157734174590187, −1.77508735537295722399337411194, 0.40793135329163387231086777620, 2.47725029048930036292738339545, 2.98947880895745698162157459260, 4.46803042865202192300336250430, 5.43386871529291280421038336886, 6.46096257472179348005206565673, 7.29359291764942009213704411073, 8.195916631883035338033675884390, 8.966229669599397263153261339282, 9.552004071982663566958208512763

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