Properties

Label 2-912-57.50-c1-0-0
Degree $2$
Conductor $912$
Sign $-0.999 + 0.0416i$
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  + (0.724 + 1.57i)3-s + (1.22 − 0.707i)5-s − 4.44·7-s + (−1.94 + 2.28i)9-s + 0.317i·11-s + (−3 − 1.73i)13-s + (2 + 1.41i)15-s + (−5.44 + 3.14i)17-s + (4.17 + 1.25i)19-s + (−3.22 − 6.99i)21-s + (−6.12 − 3.53i)23-s + (−1.50 + 2.59i)25-s + (−5.00 − 1.41i)27-s + (−1.22 + 2.12i)29-s + 4.24i·31-s + ⋯
L(s)  = 1  + (0.418 + 0.908i)3-s + (0.547 − 0.316i)5-s − 1.68·7-s + (−0.649 + 0.760i)9-s + 0.0958i·11-s + (−0.832 − 0.480i)13-s + (0.516 + 0.365i)15-s + (−1.32 + 0.763i)17-s + (0.957 + 0.287i)19-s + (−0.703 − 1.52i)21-s + (−1.27 − 0.737i)23-s + (−0.300 + 0.519i)25-s + (−0.962 − 0.272i)27-s + (−0.227 + 0.393i)29-s + 0.762i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.999 + 0.0416i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.999 + 0.0416i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0105391 - 0.505349i\)
\(L(\frac12)\) \(\approx\) \(0.0105391 - 0.505349i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.724 - 1.57i)T \)
19 \( 1 + (-4.17 - 1.25i)T \)
good5 \( 1 + (-1.22 + 0.707i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 4.44T + 7T^{2} \)
11 \( 1 - 0.317iT - 11T^{2} \)
13 \( 1 + (3 + 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.44 - 3.14i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (6.12 + 3.53i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.22 - 2.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 - 0.778iT - 37T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.449 - 0.778i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.57 - 3.21i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.550 - 0.953i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.27 + 5.67i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.22 + 5.58i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.17 - 2.98i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.39 - 9.35i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.34 - 4.24i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.1iT - 83T^{2} \)
89 \( 1 + (-8.44 + 14.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (11.8 - 6.84i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.15909648683292659280471245389, −9.741113917717187412520386129941, −9.110228961522847728107398517638, −8.232981432688085223230076809454, −7.05780558869616356742417207119, −6.08373015508812769471420596752, −5.29630988239453124757859994076, −4.14948603731631495642795911622, −3.25036604926584370314173885763, −2.24369966125123557714220916792, 0.20206774888494465385955466473, 2.17816461713731591660597698999, 2.86463956887947738533774468744, 4.02016800263063226700447286197, 5.63597412495698544717602565187, 6.40990099497161945904219905037, 6.99511427734581641421438442056, 7.77788215036734878116400313926, 9.131558782227224322317524808968, 9.480517604582193849738075651073

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