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Properties

Label 2-912-76.27-c1-0-8
Degree $2$
Conductor $912$
Sign $0.975 + 0.220i$
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  + (0.5 − 0.866i)3-s + (−0.353 + 0.612i)5-s − 0.0924i·7-s + (−0.499 − 0.866i)9-s + 4.18i·11-s + (3.54 − 2.04i)13-s + (0.353 + 0.612i)15-s + (1.22 − 2.12i)17-s + (4.26 − 0.879i)19-s + (−0.0800 − 0.0462i)21-s + (−3.10 + 1.79i)23-s + (2.24 + 3.89i)25-s − 0.999·27-s + (7.24 − 4.18i)29-s + 3.79·31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.158 + 0.273i)5-s − 0.0349i·7-s + (−0.166 − 0.288i)9-s + 1.26i·11-s + (0.982 − 0.567i)13-s + (0.0912 + 0.158i)15-s + (0.297 − 0.515i)17-s + (0.979 − 0.201i)19-s + (−0.0174 − 0.0100i)21-s + (−0.647 + 0.373i)23-s + (0.449 + 0.779i)25-s − 0.192·27-s + (1.34 − 0.776i)29-s + 0.681·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.975 + 0.220i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.975 + 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78805 - 0.199769i\)
\(L(\frac12)\) \(\approx\) \(1.78805 - 0.199769i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-4.26 + 0.879i)T \)
good5 \( 1 + (0.353 - 0.612i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.0924iT - 7T^{2} \)
11 \( 1 - 4.18iT - 11T^{2} \)
13 \( 1 + (-3.54 + 2.04i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.22 + 2.12i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.10 - 1.79i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.24 + 4.18i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.79T + 31T^{2} \)
37 \( 1 + 1.91iT - 37T^{2} \)
41 \( 1 + (-6.72 - 3.88i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.76 + 4.48i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.64 + 1.52i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.58 - 0.912i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.91 - 5.05i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.70 + 9.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.334 - 0.579i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.93 - 3.34i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.20 - 2.09i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.60 + 9.70i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.7iT - 83T^{2} \)
89 \( 1 + (-4.77 + 2.75i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.0 - 6.37i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.983397133088771774189644858721, −9.283141057856788203063300828841, −8.212505266420249761695075076982, −7.53689313879380899195850755684, −6.80518263393129476003994427652, −5.81560059384569906637580835967, −4.73518911141284951664438886890, −3.55466411528318690013530996280, −2.55474003888166128266468263951, −1.15575241465002928476298032311, 1.13093230717797165813850324578, 2.86470306940474261267282452721, 3.75456149066034382752091150134, 4.69778626450058453311245854707, 5.82096521874423665553766289379, 6.52313499577098659311102325908, 7.911259345427044130906784646698, 8.517248633339666649743526617215, 9.116133402915643666669266468482, 10.22389879787056662403930694079

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