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Properties

Label 2-912-19.5-c1-0-4
Degree $2$
Conductor $912$
Sign $0.922 - 0.385i$
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  + (−0.173 + 0.984i)3-s + (−2.97 − 2.49i)5-s + (0.613 + 1.06i)7-s + (−0.939 − 0.342i)9-s + (−1.06 + 1.83i)11-s + (0.0851 + 0.482i)13-s + (2.97 − 2.49i)15-s + (5.19 − 1.89i)17-s + (−2.77 + 3.35i)19-s + (−1.15 + 0.419i)21-s + (6.85 − 5.74i)23-s + (1.74 + 9.89i)25-s + (0.5 − 0.866i)27-s + (7.96 + 2.89i)29-s + (1.20 + 2.08i)31-s + ⋯
L(s)  = 1  + (−0.100 + 0.568i)3-s + (−1.32 − 1.11i)5-s + (0.231 + 0.401i)7-s + (−0.313 − 0.114i)9-s + (−0.319 + 0.553i)11-s + (0.0236 + 0.133i)13-s + (0.767 − 0.643i)15-s + (1.26 − 0.458i)17-s + (−0.637 + 0.770i)19-s + (−0.251 + 0.0915i)21-s + (1.42 − 1.19i)23-s + (0.349 + 1.97i)25-s + (0.0962 − 0.166i)27-s + (1.47 + 0.538i)29-s + (0.216 + 0.375i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.922 - 0.385i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.922 - 0.385i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13901 + 0.228163i\)
\(L(\frac12)\) \(\approx\) \(1.13901 + 0.228163i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (2.77 - 3.35i)T \)
good5 \( 1 + (2.97 + 2.49i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-0.613 - 1.06i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.06 - 1.83i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.0851 - 0.482i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-5.19 + 1.89i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-6.85 + 5.74i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-7.96 - 2.89i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-1.20 - 2.08i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.69T + 37T^{2} \)
41 \( 1 + (-0.277 + 1.57i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-5.08 - 4.26i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-2.03 - 0.742i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-6.80 + 5.71i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (10.7 - 3.90i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.0320 + 0.0269i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-4.20 - 1.53i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (2.02 + 1.69i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (2.66 - 15.1i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-0.809 + 4.58i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-6.24 - 10.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.46 - 8.32i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-13.5 + 4.91i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.18871129186376112535656803767, −9.139356273171300291067325157478, −8.478732101818936745777082038305, −7.87858556373602477074745062109, −6.84038201244398921426026053739, −5.42553419078798880474944966573, −4.77685771747394590098186716264, −4.05916370623889979562707958124, −2.86739781405008272199723728737, −0.950921838796326055413767102258, 0.810207487013068253464228158119, 2.74785897239701099826541746447, 3.48615433188105797222580412482, 4.59928256098379586633112432518, 5.89366011821450811590414960929, 6.80857047599236619442538296683, 7.61813104796608618126115702460, 7.954852479902503092450677097065, 9.047106360639968527876148029415, 10.48606949837921329491437458858

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