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Properties

Label 2-912-228.35-c1-0-5
Degree $2$
Conductor $912$
Sign $-0.991 + 0.129i$
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  + (0.592 + 1.62i)3-s + (−1.63 + 0.943i)7-s + (−2.29 + 1.92i)9-s + (−6.08 − 2.21i)13-s + (0.5 + 4.33i)19-s + (−2.50 − 2.10i)21-s + (−4.69 − 1.71i)25-s + (−4.5 − 2.59i)27-s + (−4.66 + 2.69i)31-s − 3.20·37-s − 11.2i·39-s + (12.9 + 2.27i)43-s + (−1.71 + 2.97i)49-s + (−6.75 + 3.37i)57-s + (−0.762 − 4.32i)61-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)3-s + (−0.617 + 0.356i)7-s + (−0.766 + 0.642i)9-s + (−1.68 − 0.614i)13-s + (0.114 + 0.993i)19-s + (−0.546 − 0.458i)21-s + (−0.939 − 0.342i)25-s + (−0.866 − 0.499i)27-s + (−0.838 + 0.484i)31-s − 0.527·37-s − 1.79i·39-s + (1.96 + 0.346i)43-s + (−0.245 + 0.425i)49-s + (−0.894 + 0.447i)57-s + (−0.0976 − 0.553i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.991 + 0.129i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.991 + 0.129i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0382192 - 0.586645i\)
\(L(\frac12)\) \(\approx\) \(0.0382192 - 0.586645i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.592 - 1.62i)T \)
19 \( 1 + (-0.5 - 4.33i)T \)
good5 \( 1 + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (1.63 - 0.943i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.08 + 2.21i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (4.66 - 2.69i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.20T + 37T^{2} \)
41 \( 1 + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (-12.9 - 2.27i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.762 + 4.32i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (1.42 + 1.69i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (14.4 - 5.26i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-5.09 - 14.0i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (-14.5 - 12.2i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.24718679862806874909949841005, −9.754464857520336809780178485325, −9.086141206289088918863128245219, −8.047565852874617761169052278490, −7.33501768240215670776393416560, −5.97257484388506737716031696924, −5.27714447209293244766160999233, −4.24415784551887976589124682038, −3.22067351513951022522462117569, −2.29949163460856591055012187923, 0.24273987949443365501497407751, 1.98685281834331396081425448479, 2.93250186900733151166076780212, 4.15054030063678758892150970109, 5.38005458653256189758530843775, 6.42098943859059962523470384971, 7.29401551983485577418727911055, 7.60113347845849298742209422136, 8.993333559508161513312304742850, 9.422531870765813854137062392490

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