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Properties

Label 2-912-57.56-c1-0-13
Degree $2$
Conductor $912$
Sign $-0.108 - 0.994i$
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  + (0.729 + 1.57i)3-s + 1.22i·5-s + 2.80·7-s + (−1.93 + 2.29i)9-s − 2.49i·11-s + 1.86i·13-s + (−1.92 + 0.894i)15-s + 4.88i·17-s + (4.12 + 1.39i)19-s + (2.05 + 4.41i)21-s − 0.424i·23-s + 3.49·25-s + (−5.01 − 1.36i)27-s − 3.86·29-s − 0.514i·31-s + ⋯
L(s)  = 1  + (0.421 + 0.906i)3-s + 0.547i·5-s + 1.06·7-s + (−0.644 + 0.764i)9-s − 0.753i·11-s + 0.518i·13-s + (−0.496 + 0.230i)15-s + 1.18i·17-s + (0.947 + 0.320i)19-s + (0.447 + 0.962i)21-s − 0.0884i·23-s + 0.699·25-s + (−0.964 − 0.262i)27-s − 0.718·29-s − 0.0924i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.108 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.108 - 0.994i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.108 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29895 + 1.44844i\)
\(L(\frac12)\) \(\approx\) \(1.29895 + 1.44844i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.729 - 1.57i)T \)
19 \( 1 + (-4.12 - 1.39i)T \)
good5 \( 1 - 1.22iT - 5T^{2} \)
7 \( 1 - 2.80T + 7T^{2} \)
11 \( 1 + 2.49iT - 11T^{2} \)
13 \( 1 - 1.86iT - 13T^{2} \)
17 \( 1 - 4.88iT - 17T^{2} \)
23 \( 1 + 0.424iT - 23T^{2} \)
29 \( 1 + 3.86T + 29T^{2} \)
31 \( 1 + 0.514iT - 31T^{2} \)
37 \( 1 - 5.16iT - 37T^{2} \)
41 \( 1 + 2.40T + 41T^{2} \)
43 \( 1 + 0.420T + 43T^{2} \)
47 \( 1 + 10.7iT - 47T^{2} \)
53 \( 1 + 7.71T + 53T^{2} \)
59 \( 1 + 3.63T + 59T^{2} \)
61 \( 1 + 1.85T + 61T^{2} \)
67 \( 1 - 14.1iT - 67T^{2} \)
71 \( 1 - 15.5T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 - 1.61iT - 79T^{2} \)
83 \( 1 - 5.68iT - 83T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 - 2.48iT - 97T^{2} \)
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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

−10.32989263167574581633678613564, −9.556883721934606307539884975002, −8.487982976367623473725400427631, −8.138721873819119114242727193038, −6.98157924779860436572420774344, −5.80450815439588025540741546252, −4.97676261654418291287813128974, −3.95091757312607156697314248257, −3.09943211819160921456116907504, −1.76374847081236126400185535220, 0.947848502445724987094186728531, 2.06767602408533053114108521055, 3.24268220566597406072366448120, 4.73747551897173093274167707180, 5.34262962185386072107773638543, 6.62448434227483034506618025032, 7.62132476136435741820796919028, 7.898355189190081011224830758162, 9.071055678994760819931164461026, 9.513864111779217403621661754923

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