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Properties

Label 2-912-76.31-c1-0-3
Degree $2$
Conductor $912$
Sign $-0.305 - 0.952i$
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 + (1 + 1.73i)5-s − 1.09i·7-s + (−0.499 + 0.866i)9-s + 5.65i·11-s + (−0.949 − 0.548i)13-s + (−0.999 + 1.73i)15-s + (1 + 1.73i)17-s + (−4 + 1.73i)19-s + (0.949 − 0.548i)21-s + (−4.89 − 2.82i)23-s + (0.500 − 0.866i)25-s − 0.999·27-s + (7.89 + 4.56i)29-s + 1.89·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.447 + 0.774i)5-s − 0.414i·7-s + (−0.166 + 0.288i)9-s + 1.70i·11-s + (−0.263 − 0.152i)13-s + (−0.258 + 0.447i)15-s + (0.242 + 0.420i)17-s + (−0.917 + 0.397i)19-s + (0.207 − 0.119i)21-s + (−1.02 − 0.589i)23-s + (0.100 − 0.173i)25-s − 0.192·27-s + (1.46 + 0.846i)29-s + 0.341·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.305 - 0.952i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.305 - 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.959886 + 1.31643i\)
\(L(\frac12)\) \(\approx\) \(0.959886 + 1.31643i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4 - 1.73i)T \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 1.09iT - 7T^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 + (0.949 + 0.548i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (4.89 + 2.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.89 - 4.56i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.89T + 31T^{2} \)
37 \( 1 - 4.56iT - 37T^{2} \)
41 \( 1 + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.5 - 2.59i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.89 - 1.09i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.10 - 0.635i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.89 + 6.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0505 + 0.0874i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.39 - 9.35i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2 + 3.46i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.39 + 11.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.94 - 8.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 + (-7.89 - 4.56i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (15.7 - 9.12i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.17033795475653906362016479350, −9.918725935561531063763449616709, −8.681684126552928986263040167111, −7.82737605315604474011538907457, −6.87411696797207155060374021598, −6.20840639007191476755932148055, −4.82811945408777360472846152188, −4.16739531584109114349256442441, −2.88259990907307114092747080096, −1.90064665313956650674594654492, 0.73883020638650200280207645316, 2.15108278641234858075123492236, 3.24546484508489120488251313757, 4.54595254351110118708602977240, 5.69395721313793248727688499261, 6.18231425834640392058512343031, 7.38487442109051200470836847268, 8.484529176716994991803707722940, 8.721601541338189706372577412211, 9.679812841651849144197755297460

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