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Properties

Label 2-912-19.12-c2-0-26
Degree $2$
Conductor $912$
Sign $-0.117 + 0.993i$
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 + (1.57 + 2.72i)5-s − 10.4·7-s + (1.5 − 2.59i)9-s − 5.29·11-s + (1.29 + 0.749i)13-s + (4.72 + 2.72i)15-s + (0.157 + 0.272i)17-s + (16.2 + 9.82i)19-s + (−15.6 + 9.06i)21-s + (22.1 − 38.3i)23-s + (7.53 − 13.0i)25-s − 5.19i·27-s + (−35.7 − 20.6i)29-s − 21.8i·31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (0.315 + 0.545i)5-s − 1.49·7-s + (0.166 − 0.288i)9-s − 0.481·11-s + (0.0998 + 0.0576i)13-s + (0.315 + 0.181i)15-s + (0.00927 + 0.0160i)17-s + (0.855 + 0.517i)19-s + (−0.747 + 0.431i)21-s + (0.962 − 1.66i)23-s + (0.301 − 0.522i)25-s − 0.192i·27-s + (−1.23 − 0.711i)29-s − 0.703i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.117 + 0.993i$
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: \((2,\ 912,\ (\ :1),\ -0.117 + 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.354539812\)
\(L(\frac12)\) \(\approx\) \(1.354539812\)
\(L(2)\) 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 \( 1 + (-16.2 - 9.82i)T \)
good5 \( 1 + (-1.57 - 2.72i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 10.4T + 49T^{2} \)
11 \( 1 + 5.29T + 121T^{2} \)
13 \( 1 + (-1.29 - 0.749i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-0.157 - 0.272i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-22.1 + 38.3i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (35.7 + 20.6i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 21.8iT - 961T^{2} \)
37 \( 1 + 60.4iT - 1.36e3T^{2} \)
41 \( 1 + (18.6 - 10.7i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (6.38 + 11.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-11.3 + 19.6i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (30.3 + 17.5i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (6.95 - 4.01i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-14.3 + 24.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-38.9 - 22.4i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-64.8 + 37.4i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (10.2 + 17.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (112. - 65.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 78.5T + 6.88e3T^{2} \)
89 \( 1 + (-131. - 75.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-17.9 + 10.3i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.680689647952949027448531516856, −8.969877561421868969584394639733, −7.927273053355132494288642515138, −7.00588610649556721360750208020, −6.39811332916779038353679943986, −5.49534892447161086429068988524, −3.99005391965537935546382894614, −3.05569902283638937422739102530, −2.28643614050130984803927206190, −0.41668407847931583737825067116, 1.33962036513610666581391438460, 2.99239300983070621370807067754, 3.46820502364884643654073746483, 4.95258934102186738575128243707, 5.62486419349735283628716846168, 6.82761967067300293080336834326, 7.52089600876511747454073550477, 8.722320883497129727188233844871, 9.395292634096619857855844249849, 9.791171548477927418788754472161

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