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Properties

Label 2-912-19.12-c2-0-21
Degree $2$
Conductor $912$
Sign $0.302 + 0.953i$
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 + (−4.28 − 7.41i)5-s + 10.9·7-s + (1.5 − 2.59i)9-s − 6.12·11-s + (3.96 + 2.28i)13-s + (12.8 + 7.41i)15-s + (11.9 + 20.7i)17-s + (18.8 + 2.24i)19-s + (−16.4 + 9.49i)21-s + (20.6 − 35.7i)23-s + (−24.1 + 41.8i)25-s + 5.19i·27-s + (−15.3 − 8.85i)29-s + 0.0804i·31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (−0.856 − 1.48i)5-s + 1.56·7-s + (0.166 − 0.288i)9-s − 0.556·11-s + (0.305 + 0.176i)13-s + (0.856 + 0.494i)15-s + (0.705 + 1.22i)17-s + (0.993 + 0.117i)19-s + (−0.783 + 0.452i)21-s + (0.897 − 1.55i)23-s + (−0.966 + 1.67i)25-s + 0.192i·27-s + (−0.529 − 0.305i)29-s + 0.00259i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.302 + 0.953i$
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: \((2,\ 912,\ (\ :1),\ 0.302 + 0.953i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.537463653\)
\(L(\frac12)\) \(\approx\) \(1.537463653\)
\(L(2)\) 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 \( 1 + (-18.8 - 2.24i)T \)
good5 \( 1 + (4.28 + 7.41i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 10.9T + 49T^{2} \)
11 \( 1 + 6.12T + 121T^{2} \)
13 \( 1 + (-3.96 - 2.28i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-11.9 - 20.7i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-20.6 + 35.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (15.3 + 8.85i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 0.0804iT - 961T^{2} \)
37 \( 1 + 19.7iT - 1.36e3T^{2} \)
41 \( 1 + (1.52 - 0.882i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-22.4 - 38.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-6.58 + 11.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (79.1 + 45.7i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (1.30 - 0.755i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-39.8 + 69.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (56.2 + 32.4i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-48.2 + 27.8i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (23.6 + 41.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-107. + 61.9i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 66.8T + 6.88e3T^{2} \)
89 \( 1 + (-35.8 - 20.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (24.2 - 14.0i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.665450470311021990698687558909, −8.668562269858697039298864502708, −8.129679075746829707584948037839, −7.53217374305102090450111197342, −5.98705306510180760329526494899, −4.98472812296462950673889666394, −4.69380581242902256993549431758, −3.64315607769040408797198223982, −1.66613945366018835564168758612, −0.66128202185567200025227352993, 1.14756411072265600788322245985, 2.66217569338724242785865234042, 3.61007841181481608383496550622, 4.95377974903871690545303920695, 5.58043663435091926562211677664, 6.94946553104154951357293823051, 7.59948969427686616202888019431, 7.86084228131539844444462762308, 9.269221016622454293537786622804, 10.43076099001720355307294880644

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