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Properties

Label 2-912-19.6-c1-0-19
Degree $2$
Conductor $912$
Sign $-0.962 + 0.270i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 − 0.342i)3-s + (−0.613 − 3.47i)5-s + (1.85 − 3.21i)7-s + (0.766 + 0.642i)9-s + (−2.64 − 4.58i)11-s + (0.213 − 0.0775i)13-s + (−0.613 + 3.47i)15-s + (−1.26 + 1.06i)17-s + (4.17 − 1.24i)19-s + (−2.84 + 2.38i)21-s + (−1.50 + 8.54i)23-s + (−7.02 + 2.55i)25-s + (−0.500 − 0.866i)27-s + (0.0923 + 0.0775i)29-s + (1.56 − 2.70i)31-s + ⋯
L(s)  = 1  + (−0.542 − 0.197i)3-s + (−0.274 − 1.55i)5-s + (0.702 − 1.21i)7-s + (0.255 + 0.214i)9-s + (−0.797 − 1.38i)11-s + (0.0590 − 0.0215i)13-s + (−0.158 + 0.898i)15-s + (−0.307 + 0.257i)17-s + (0.958 − 0.285i)19-s + (−0.621 + 0.521i)21-s + (−0.314 + 1.78i)23-s + (−1.40 + 0.511i)25-s + (−0.0962 − 0.166i)27-s + (0.0171 + 0.0143i)29-s + (0.280 − 0.485i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.962 + 0.270i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.962 + 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.140763 - 1.01975i\)
\(L(\frac12)\) \(\approx\) \(0.140763 - 1.01975i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (-4.17 + 1.24i)T \)
good5 \( 1 + (0.613 + 3.47i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-1.85 + 3.21i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.64 + 4.58i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.213 + 0.0775i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (1.26 - 1.06i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (1.50 - 8.54i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-0.0923 - 0.0775i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.56 + 2.70i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.12T + 37T^{2} \)
41 \( 1 + (6.67 + 2.43i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.929 - 5.27i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-1.92 - 1.61i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-1.03 + 5.84i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (0.167 - 0.140i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.273 - 1.55i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (11.8 + 9.95i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.235 - 1.33i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (2.27 + 0.829i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (2.69 + 0.979i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-0.960 + 1.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-11.4 + 4.15i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-13.4 + 11.2i)T + (16.8 - 95.5i)T^{2} \)
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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

−9.716064658931575350992546520528, −8.748540458396870150856975339690, −7.893974529549544641320788230953, −7.50237669027401671442885132544, −6.00825999312891052057371409912, −5.23448674314607879384026985247, −4.52941278077148615100382150434, −3.48860813624675571914527641771, −1.43980836266742677590371950763, −0.55231471161788936492440829642, 2.15532066579000429664137820055, 2.90602185090911719357834615165, 4.38866201462903379860327832906, 5.23443945525094831282179516213, 6.21302080102935703998721174192, 7.05754969685970155997017654478, 7.77609546790106980615300664417, 8.798467409544410252634334668383, 9.979488893733830774039269085750, 10.43212057727927763624647511763

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