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Properties

Label 2-912-19.12-c2-0-31
Degree $2$
Conductor $912$
Sign $-0.471 + 0.882i$
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 + (0.113 + 0.197i)5-s + 2.53·7-s + (1.5 − 2.59i)9-s − 11.3·11-s + (−8.88 − 5.12i)13-s + (0.341 + 0.197i)15-s + (8.13 + 14.0i)17-s + (−11.5 − 15.0i)19-s + (3.80 − 2.19i)21-s + (16.8 − 29.1i)23-s + (12.4 − 21.6i)25-s − 5.19i·27-s + (−38.9 − 22.4i)29-s − 46.5i·31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (0.0227 + 0.0394i)5-s + 0.362·7-s + (0.166 − 0.288i)9-s − 1.03·11-s + (−0.683 − 0.394i)13-s + (0.0227 + 0.0131i)15-s + (0.478 + 0.828i)17-s + (−0.608 − 0.793i)19-s + (0.181 − 0.104i)21-s + (0.731 − 1.26i)23-s + (0.498 − 0.864i)25-s − 0.192i·27-s + (−1.34 − 0.775i)29-s − 1.50i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 + 0.882i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.471 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.471 + 0.882i$
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: \((2,\ 912,\ (\ :1),\ -0.471 + 0.882i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.456128101\)
\(L(\frac12)\) \(\approx\) \(1.456128101\)
\(L(2)\) 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 \( 1 + (11.5 + 15.0i)T \)
good5 \( 1 + (-0.113 - 0.197i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 2.53T + 49T^{2} \)
11 \( 1 + 11.3T + 121T^{2} \)
13 \( 1 + (8.88 + 5.12i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-8.13 - 14.0i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-16.8 + 29.1i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (38.9 + 22.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 46.5iT - 961T^{2} \)
37 \( 1 - 72.5iT - 1.36e3T^{2} \)
41 \( 1 + (-17.8 + 10.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (28.2 + 48.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (10.5 - 18.3i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-49.1 - 28.3i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-91.4 + 52.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-7.75 + 13.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (90.4 + 52.2i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-16.0 + 9.28i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-17.3 - 30.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-53.8 + 31.1i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 112.T + 6.88e3T^{2} \)
89 \( 1 + (64.7 + 37.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (63.2 - 36.5i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.694172663913387931914586589978, −8.517655580822813134349951576685, −8.078605301889285640404534120507, −7.19851494503862272036996274981, −6.23712110828076749459640344574, −5.15382370745469479488022691466, −4.25928216531447939990996013355, −2.89508854211349376731138894249, −2.11762456954336465991371789503, −0.41950560484311375390703480388, 1.58365519167499975195112812521, 2.78938722526085578511397397309, 3.75998826433945631242710386754, 5.04878532840746112903589749300, 5.47708837241375440770539895429, 7.13945395962771102496883389008, 7.53075508525581502644711188073, 8.583975064780224244742498052902, 9.324496438053467212756592098805, 10.08656971344690969206804802136

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