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Properties

Label 2-912-57.8-c1-0-9
Degree $2$
Conductor $912$
Sign $-0.823 - 0.567i$
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  + (−0.5 + 1.65i)3-s + (−0.686 − 0.396i)5-s + 2.37·7-s + (−2.5 − 1.65i)9-s + 3.46i·11-s + (−2.87 + 1.65i)13-s + (1 − 0.939i)15-s + (−0.686 − 0.396i)17-s + (4 + 1.73i)19-s + (−1.18 + 3.93i)21-s + (−6.43 + 3.71i)23-s + (−2.18 − 3.78i)25-s + (4 − 3.31i)27-s + (2.68 + 4.65i)29-s + 4.40i·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.957i)3-s + (−0.306 − 0.177i)5-s + 0.896·7-s + (−0.833 − 0.552i)9-s + 1.04i·11-s + (−0.796 + 0.459i)13-s + (0.258 − 0.242i)15-s + (−0.166 − 0.0960i)17-s + (0.917 + 0.397i)19-s + (−0.258 + 0.858i)21-s + (−1.34 + 0.774i)23-s + (−0.437 − 0.757i)25-s + (0.769 − 0.638i)27-s + (0.498 + 0.863i)29-s + 0.790i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.823 - 0.567i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.823 - 0.567i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.283295 + 0.910030i\)
\(L(\frac12)\) \(\approx\) \(0.283295 + 0.910030i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 1.65i)T \)
19 \( 1 + (-4 - 1.73i)T \)
good5 \( 1 + (0.686 + 0.396i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 2.37T + 7T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 + (2.87 - 1.65i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.686 + 0.396i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (6.43 - 3.71i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.68 - 4.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.40iT - 31T^{2} \)
37 \( 1 - 7.86iT - 37T^{2} \)
41 \( 1 + (-0.313 + 0.543i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.127 + 0.221i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.68 - 2.12i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.68 + 9.84i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.68 - 6.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.2 - 5.91i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.31 - 5.73i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.87 + 4.97i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.98 + 5.76i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + (3.68 + 6.38i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.0 - 6.38i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.09844246167211004549723694195, −9.921999110421979644024384359266, −8.820184600687354982025649189542, −7.980190605196201310471680812654, −7.12881555869283679029830742699, −5.92529115743043858493509836800, −4.80223509625920338996651104558, −4.53952548738289254363154001466, −3.26969318866354942071206103045, −1.76467755419952867882143591227, 0.45964583906617574729050373151, 1.96259242993165211553045965461, 3.08600351985784616355767759746, 4.50325969159922812353089189089, 5.56020265099271106505085623521, 6.23202321218768358836354665819, 7.48214912366282511964832902356, 7.83972739186757612834665163419, 8.641027564854350877512084170453, 9.790513712379631971653102267781

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