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Properties

Label 2-912-76.67-c1-0-4
Degree $2$
Conductor $912$
Sign $0.418 - 0.908i$
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  + (0.766 − 0.642i)3-s + (−3.87 − 1.41i)5-s + (3.15 + 1.82i)7-s + (0.173 − 0.984i)9-s + (−3.84 + 2.21i)11-s + (−1.63 + 1.95i)13-s + (−3.87 + 1.41i)15-s + (0.911 + 5.17i)17-s + (3.02 + 3.14i)19-s + (3.59 − 0.633i)21-s + (0.131 + 0.361i)23-s + (9.19 + 7.71i)25-s + (−0.500 − 0.866i)27-s + (1.63 + 0.287i)29-s + (3.82 − 6.62i)31-s + ⋯
L(s)  = 1  + (0.442 − 0.371i)3-s + (−1.73 − 0.630i)5-s + (1.19 + 0.689i)7-s + (0.0578 − 0.328i)9-s + (−1.15 + 0.669i)11-s + (−0.454 + 0.541i)13-s + (−1.00 + 0.364i)15-s + (0.221 + 1.25i)17-s + (0.693 + 0.720i)19-s + (0.783 − 0.138i)21-s + (0.0274 + 0.0753i)23-s + (1.83 + 1.54i)25-s + (−0.0962 − 0.166i)27-s + (0.303 + 0.0534i)29-s + (0.687 − 1.19i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.418 - 0.908i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.418 - 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.934025 + 0.597735i\)
\(L(\frac12)\) \(\approx\) \(0.934025 + 0.597735i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (-3.02 - 3.14i)T \)
good5 \( 1 + (3.87 + 1.41i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-3.15 - 1.82i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.84 - 2.21i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.63 - 1.95i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.911 - 5.17i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-0.131 - 0.361i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-1.63 - 0.287i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (-3.82 + 6.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.69iT - 37T^{2} \)
41 \( 1 + (-6.15 - 7.33i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.24 - 3.42i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (6.65 + 1.17i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-3.78 - 10.3i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-0.705 - 3.99i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (8.72 - 3.17i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.516 - 2.92i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (4.17 + 1.52i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (7.13 - 5.98i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (11.3 - 9.55i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-8.35 - 4.82i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.58 + 10.2i)T + (-15.4 - 87.6i)T^{2} \)
97 \( 1 + (4.55 - 0.803i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.27824158705455719377098612372, −9.136119761029588234951054713510, −8.197717788387618567869785475849, −7.925251110666863177627514443361, −7.33742070757841061247916582373, −5.77946440829965948868370676613, −4.71307525150881225844695610360, −4.13693470681974073010482326473, −2.75942131686233851857247573555, −1.45307702660683301898182348307, 0.53213206608217611271676249825, 2.79119404060808317469800085422, 3.41539650359525545212988227525, 4.67194739753127124213157942448, 5.08068378175114068116618880862, 6.95037077049840313661389840415, 7.69240981755011401106058794967, 7.962731882765716073866563109408, 8.867736681687785060805651451285, 10.27845068625987784030580144447

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