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Properties

Label 2-912-19.12-c2-0-18
Degree $2$
Conductor $912$
Sign $0.797 + 0.602i$
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 + (−3.25 − 5.64i)5-s + 11.0·7-s + (1.5 − 2.59i)9-s + 17.2·11-s + (13.2 + 7.66i)13-s + (9.77 + 5.64i)15-s + (−12.0 − 20.9i)17-s + (−16.6 + 9.07i)19-s + (−16.5 + 9.53i)21-s + (−12.6 + 21.9i)23-s + (−8.74 + 15.1i)25-s + 5.19i·27-s + (10.6 + 6.17i)29-s − 15.5i·31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (−0.651 − 1.12i)5-s + 1.57·7-s + (0.166 − 0.288i)9-s + 1.56·11-s + (1.02 + 0.589i)13-s + (0.651 + 0.376i)15-s + (−0.711 − 1.23i)17-s + (−0.878 + 0.477i)19-s + (−0.786 + 0.454i)21-s + (−0.551 + 0.955i)23-s + (−0.349 + 0.605i)25-s + 0.192i·27-s + (0.368 + 0.212i)29-s − 0.502i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 + 0.602i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.797 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.797 + 0.602i$
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: \((2,\ 912,\ (\ :1),\ 0.797 + 0.602i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.885297471\)
\(L(\frac12)\) \(\approx\) \(1.885297471\)
\(L(2)\) 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 \( 1 + (16.6 - 9.07i)T \)
good5 \( 1 + (3.25 + 5.64i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 11.0T + 49T^{2} \)
11 \( 1 - 17.2T + 121T^{2} \)
13 \( 1 + (-13.2 - 7.66i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (12.0 + 20.9i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (12.6 - 21.9i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-10.6 - 6.17i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 15.5iT - 961T^{2} \)
37 \( 1 - 17.7iT - 1.36e3T^{2} \)
41 \( 1 + (-43.6 + 25.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-31.6 - 54.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-35.1 + 60.8i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (12.8 + 7.40i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-59.8 + 34.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-18.6 + 32.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-85.3 - 49.2i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (37.5 - 21.6i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (48.2 + 83.5i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (40.1 - 23.1i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 66.5T + 6.88e3T^{2} \)
89 \( 1 + (-32.5 - 18.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (5.11 - 2.95i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.634129932053223898439564054379, −8.809318556744829272890622531784, −8.392948022520674819263754649315, −7.32670053440175452604322531933, −6.27042303926118558780028374290, −5.24007909581122124246365222802, −4.28596175129995189106771268287, −4.05099472073517771901187802616, −1.77081597085415917246229203745, −0.860333751985016733787490310128, 1.10549134334923535469236117714, 2.29356575555278049449524736632, 3.91076098287910909587854658723, 4.39017206417581032211284968567, 5.91213927507058384969828999250, 6.52126406203268297988304598150, 7.37614957668691843060146763606, 8.315411206934206288955955116311, 8.820962903783325696499421292889, 10.49764868604211439693941497907

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