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Properties

Label 2-912-228.83-c1-0-0
Degree $2$
Conductor $912$
Sign $-0.777 - 0.628i$
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  + (0.325 − 1.70i)3-s + (−0.164 + 0.0950i)5-s + 0.883i·7-s + (−2.78 − 1.10i)9-s − 3.88·11-s + (−2.04 + 3.54i)13-s + (0.108 + 0.310i)15-s + (−3.97 + 2.29i)17-s + (−3.03 − 3.13i)19-s + (1.50 + 0.287i)21-s + (1.39 − 2.41i)23-s + (−2.48 + 4.29i)25-s + (−2.78 + 4.38i)27-s + (−2.80 − 1.62i)29-s − 4.38i·31-s + ⋯
L(s)  = 1  + (0.187 − 0.982i)3-s + (−0.0735 + 0.0424i)5-s + 0.333i·7-s + (−0.929 − 0.368i)9-s − 1.17·11-s + (−0.567 + 0.982i)13-s + (0.0279 + 0.0802i)15-s + (−0.963 + 0.556i)17-s + (−0.695 − 0.718i)19-s + (0.327 + 0.0626i)21-s + (0.290 − 0.503i)23-s + (−0.496 + 0.859i)25-s + (−0.536 + 0.843i)27-s + (−0.521 − 0.300i)29-s − 0.787i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.777 - 0.628i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.777 - 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000717417 + 0.00202894i\)
\(L(\frac12)\) \(\approx\) \(0.000717417 + 0.00202894i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.325 + 1.70i)T \)
19 \( 1 + (3.03 + 3.13i)T \)
good5 \( 1 + (0.164 - 0.0950i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 0.883iT - 7T^{2} \)
11 \( 1 + 3.88T + 11T^{2} \)
13 \( 1 + (2.04 - 3.54i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.97 - 2.29i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.39 + 2.41i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.80 + 1.62i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.38iT - 31T^{2} \)
37 \( 1 - 0.128T + 37T^{2} \)
41 \( 1 + (1.63 - 0.946i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.36 + 1.94i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.07 + 3.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.2 + 6.51i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.94 - 8.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.04 - 5.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.7 + 7.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.43 - 7.67i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.37 - 4.10i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.305 + 0.176i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.00T + 83T^{2} \)
89 \( 1 + (0.493 + 0.285i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.981 - 1.70i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.56495473758251822256303632514, −9.326802823674994097331262908908, −8.719760010709016529430260041157, −7.81533859604691535323359770723, −7.07728168639622287609511500197, −6.27272964724496507782601538099, −5.30195408550595928239831182163, −4.16266310470688747803794877137, −2.66273660169929984164353891139, −1.98540887856771887460173812854, 0.000893570578645070776687135647, 2.37904262069928759619932141903, 3.32783407965434925638118784001, 4.47626095911622397963179317173, 5.17351483019549195323712194115, 6.11146978365716097075097385705, 7.47800093127605804747885507037, 8.098146683198667186930850980149, 9.012708674029559392847685215417, 9.884008684102306187278995772107

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