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Properties

Label 2-912-19.4-c1-0-8
Degree $2$
Conductor $912$
Sign $0.837 - 0.546i$
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 + (1.93 − 1.62i)5-s + (−1.61 + 2.79i)7-s + (−0.939 + 0.342i)9-s + (1.55 + 2.68i)11-s + (−1.05 + 5.97i)13-s + (−1.93 − 1.62i)15-s + (5.58 + 2.03i)17-s + (−4.34 + 0.405i)19-s + (3.03 + 1.10i)21-s + (5.14 + 4.31i)23-s + (0.245 − 1.39i)25-s + (0.5 + 0.866i)27-s + (−6.69 + 2.43i)29-s + (3.81 − 6.61i)31-s + ⋯
L(s)  = 1  + (−0.100 − 0.568i)3-s + (0.867 − 0.727i)5-s + (−0.609 + 1.05i)7-s + (−0.313 + 0.114i)9-s + (0.468 + 0.811i)11-s + (−0.292 + 1.65i)13-s + (−0.500 − 0.420i)15-s + (1.35 + 0.493i)17-s + (−0.995 + 0.0929i)19-s + (0.661 + 0.240i)21-s + (1.07 + 0.900i)23-s + (0.0490 − 0.278i)25-s + (0.0962 + 0.166i)27-s + (−1.24 + 0.452i)29-s + (0.685 − 1.18i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.837 - 0.546i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.837 - 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50734 + 0.447944i\)
\(L(\frac12)\) \(\approx\) \(1.50734 + 0.447944i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.34 - 0.405i)T \)
good5 \( 1 + (-1.93 + 1.62i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (1.61 - 2.79i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.55 - 2.68i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.05 - 5.97i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-5.58 - 2.03i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-5.14 - 4.31i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (6.69 - 2.43i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-3.81 + 6.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.10T + 37T^{2} \)
41 \( 1 + (1.22 + 6.93i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-7.17 + 6.02i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (4.37 - 1.59i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (4.86 + 4.08i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (0.252 + 0.0918i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-1.21 - 1.02i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (1.22 - 0.446i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-6.34 + 5.31i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.159 - 0.907i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-2.09 - 11.8i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.0432 + 0.0748i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.22 - 6.96i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (3.12 + 1.13i)T + (74.3 + 62.3i)T^{2} \)
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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

−9.754005337228866321062506681000, −9.360188911625038434624523672108, −8.769457632181333749551643566063, −7.54014089319520716812227290992, −6.63010667549097200101514025358, −5.86467563253229696477388377449, −5.13538314088787886548295390264, −3.86992115488630051825484937009, −2.28762985587180160207075257781, −1.56767296172337965774789387403, 0.792670850154611033897023385964, 2.86994894853912012362345238755, 3.38753954003257173597165807640, 4.71225642709999125180684564740, 5.81454729037782894267472669734, 6.41681676881932127019234896337, 7.40528191126588801622487211369, 8.363687021123174028200534049881, 9.507644042190823277312495549535, 10.13766636772417691578605882050

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