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Properties

Label 2-912-57.8-c1-0-19
Degree $2$
Conductor $912$
Sign $0.992 - 0.120i$
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  + (−1.57 + 0.724i)3-s + (1.22 + 0.707i)5-s + 3.73·7-s + (1.94 − 2.28i)9-s + 0.378i·11-s + (3.23 − 1.86i)13-s + (−2.43 − 0.224i)15-s + (−4.24 − 2.44i)17-s + (1.73 − 4i)19-s + (−5.87 + 2.70i)21-s + (0.328 − 0.189i)23-s + (−1.50 − 2.59i)25-s + (−1.41 + 5.00i)27-s + (3.86 + 6.69i)29-s − 4.46i·31-s + ⋯
L(s)  = 1  + (−0.908 + 0.418i)3-s + (0.547 + 0.316i)5-s + 1.41·7-s + (0.649 − 0.760i)9-s + 0.114i·11-s + (0.896 − 0.517i)13-s + (−0.629 − 0.0580i)15-s + (−1.02 − 0.594i)17-s + (0.397 − 0.917i)19-s + (−1.28 + 0.590i)21-s + (0.0684 − 0.0395i)23-s + (−0.300 − 0.519i)25-s + (−0.272 + 0.962i)27-s + (0.717 + 1.24i)29-s − 0.801i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.992 - 0.120i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.992 - 0.120i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56128 + 0.0945517i\)
\(L(\frac12)\) \(\approx\) \(1.56128 + 0.0945517i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.57 - 0.724i)T \)
19 \( 1 + (-1.73 + 4i)T \)
good5 \( 1 + (-1.22 - 0.707i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.73T + 7T^{2} \)
11 \( 1 - 0.378iT - 11T^{2} \)
13 \( 1 + (-3.23 + 1.86i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.24 + 2.44i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.328 + 0.189i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.86 - 6.69i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.46iT - 31T^{2} \)
37 \( 1 - 4.26iT - 37T^{2} \)
41 \( 1 + (-2.82 + 4.89i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.13 + 1.96i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.14 + 5.27i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.01 - 5.22i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.19 - 7.26i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.76 - 3.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.79 - 3.10i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.06 - 1.76i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.72iT - 83T^{2} \)
89 \( 1 + (-3.67 - 6.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.46 - 3.73i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.47795277002357231408304447069, −9.283857593082758202907917426220, −8.591130768681698998860806388630, −7.42164803628552455708433342270, −6.58805722477949247409561795138, −5.62810114961104821428600656961, −4.92157684325426196081297762184, −4.06850459074469923433063127228, −2.49054356242141225989614853120, −1.05514743410002496387119168017, 1.26823089774851105314004965969, 2.03331240013732554942859836207, 4.07137618608077416496931288538, 4.88351216600612035962475044847, 5.79912757310749157967626618289, 6.41859009745905277366024000329, 7.60499432183130609746338491141, 8.283685247160205696974901373394, 9.194526012803462008835272014404, 10.30781151021026923662906190070

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