Properties

Label 2-912-19.12-c2-0-5
Degree $2$
Conductor $912$
Sign $-0.703 - 0.710i$
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 + (2.88 + 4.98i)5-s − 1.94·7-s + (1.5 − 2.59i)9-s − 8.46·11-s + (−16.7 − 9.66i)13-s + (8.64 + 4.98i)15-s + (12.5 + 21.7i)17-s + (−17.8 + 6.59i)19-s + (−2.91 + 1.68i)21-s + (−15.6 + 27.0i)23-s + (−4.09 + 7.08i)25-s − 5.19i·27-s + (13.7 + 7.91i)29-s + 14.4i·31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (0.576 + 0.997i)5-s − 0.277·7-s + (0.166 − 0.288i)9-s − 0.769·11-s + (−1.28 − 0.743i)13-s + (0.576 + 0.332i)15-s + (0.737 + 1.27i)17-s + (−0.937 + 0.347i)19-s + (−0.138 + 0.0801i)21-s + (−0.680 + 1.17i)23-s + (−0.163 + 0.283i)25-s − 0.192i·27-s + (0.472 + 0.272i)29-s + 0.465i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.703 - 0.710i$
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: \((2,\ 912,\ (\ :1),\ -0.703 - 0.710i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.109662675\)
\(L(\frac12)\) \(\approx\) \(1.109662675\)
\(L(2)\) 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 \( 1 + (17.8 - 6.59i)T \)
good5 \( 1 + (-2.88 - 4.98i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 1.94T + 49T^{2} \)
11 \( 1 + 8.46T + 121T^{2} \)
13 \( 1 + (16.7 + 9.66i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-12.5 - 21.7i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (15.6 - 27.0i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-13.7 - 7.91i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 14.4iT - 961T^{2} \)
37 \( 1 - 41.6iT - 1.36e3T^{2} \)
41 \( 1 + (-1.53 + 0.885i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (14.4 + 25.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (1.70 - 2.95i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (80.0 + 46.2i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (4.27 - 2.46i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-7.45 + 12.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-70.4 - 40.6i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (52.9 - 30.5i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (38.8 + 67.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (84.0 - 48.5i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 102.T + 6.88e3T^{2} \)
89 \( 1 + (-103. - 59.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (42.1 - 24.3i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.18914759596574239518531009326, −9.650089799111214576064723364651, −8.274940724535651606715008251331, −7.78983665192387102625655854464, −6.78687829231156662868883857232, −6.04701139995043785345587372169, −5.05544512933246073031641130515, −3.55810158920937104382442766508, −2.77376167965285578524015071376, −1.79487472944977619141785993823, 0.30133813315200498856104728754, 2.03102092744390129974719414426, 2.88512413899760675762133594860, 4.49126151420481762746888035930, 4.90332605524740025645185182422, 6.01069726673314574862225674915, 7.15055977365603238494186334421, 8.013626265275127600267178245016, 8.893465266223731123068586382150, 9.608712938308747656036279764597

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