Properties

Label 2-912-19.12-c2-0-16
Degree $2$
Conductor $912$
Sign $0.926 - 0.376i$
Analytic cond. $24.8502$
Root an. cond. $4.98499$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 − 0.866i)3-s + (2.60 + 4.51i)5-s + 5.77·7-s + (1.5 − 2.59i)9-s + 14.2·11-s + (1.04 + 0.605i)13-s + (7.82 + 4.51i)15-s + (−0.848 − 1.46i)17-s + (−0.755 + 18.9i)19-s + (8.66 − 5.00i)21-s + (1.02 − 1.77i)23-s + (−1.09 + 1.89i)25-s − 5.19i·27-s + (−15.6 − 9.02i)29-s − 4.61i·31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (0.521 + 0.903i)5-s + 0.825·7-s + (0.166 − 0.288i)9-s + 1.29·11-s + (0.0806 + 0.0465i)13-s + (0.521 + 0.301i)15-s + (−0.0499 − 0.0864i)17-s + (−0.0397 + 0.999i)19-s + (0.412 − 0.238i)21-s + (0.0445 − 0.0772i)23-s + (−0.0436 + 0.0756i)25-s − 0.192i·27-s + (−0.538 − 0.311i)29-s − 0.148i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.926 - 0.376i$
Analytic conductor: \(24.8502\)
Root analytic conductor: \(4.98499\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1),\ 0.926 - 0.376i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.021401460\)
\(L(\frac12)\) \(\approx\) \(3.021401460\)
\(L(2)\) 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 \( 1 + (-1.5 + 0.866i)T \)
19 \( 1 + (0.755 - 18.9i)T \)
good5 \( 1 + (-2.60 - 4.51i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 5.77T + 49T^{2} \)
11 \( 1 - 14.2T + 121T^{2} \)
13 \( 1 + (-1.04 - 0.605i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (0.848 + 1.46i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-1.02 + 1.77i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (15.6 + 9.02i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 4.61iT - 961T^{2} \)
37 \( 1 + 20.1iT - 1.36e3T^{2} \)
41 \( 1 + (-7.14 + 4.12i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-37.2 - 64.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (0.118 - 0.205i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (38.2 + 22.0i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-23.8 + 13.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (33.0 - 57.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-89.0 - 51.4i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (88.2 - 50.9i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (25.0 + 43.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-134. + 77.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 9.99T + 6.88e3T^{2} \)
89 \( 1 + (-103. - 59.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (0.945 - 0.546i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.853139909126200929810104834144, −9.159232201374987325831243381516, −8.212011144429207157887522443323, −7.43049709851197203829227667683, −6.52443507202213712282861377157, −5.85382093741847229355013200628, −4.45094488558569275528728277564, −3.49796917186519104105597825283, −2.30253143528226635870388585050, −1.35508306687755303782302583662, 1.10284278255416948959813303264, 2.06997257068353461957602547380, 3.55890901551104713486305238851, 4.58069746447362537098836573340, 5.23543780677362667986982800654, 6.38376073307935287754294669611, 7.41574179579207533897631259876, 8.419673434135046250970080192775, 9.080809945922919309708175900589, 9.493474420610894183491880213186

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