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Properties

Label 2-912-57.56-c1-0-18
Degree $2$
Conductor $912$
Sign $0.662 - 0.749i$
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 + 1.41i)3-s − 1.41i·5-s + 4·7-s + (−1.00 + 2.82i)9-s + 5.65i·11-s − 4.24i·13-s + (2.00 − 1.41i)15-s + 2.82i·17-s + (−1 − 4.24i)19-s + (4 + 5.65i)21-s + 1.41i·23-s + 2.99·25-s + (−5.00 + 1.41i)27-s + 6·29-s + 4.24i·31-s + ⋯
L(s)  = 1  + (0.577 + 0.816i)3-s − 0.632i·5-s + 1.51·7-s + (−0.333 + 0.942i)9-s + 1.70i·11-s − 1.17i·13-s + (0.516 − 0.365i)15-s + 0.685i·17-s + (−0.229 − 0.973i)19-s + (0.872 + 1.23i)21-s + 0.294i·23-s + 0.599·25-s + (−0.962 + 0.272i)27-s + 1.11·29-s + 0.762i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.662 - 0.749i$
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: \((2,\ 912,\ (\ :1/2),\ 0.662 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02621 + 0.913319i\)
\(L(\frac12)\) \(\approx\) \(2.02621 + 0.913319i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.41i)T \)
19 \( 1 + (1 + 4.24i)T \)
good5 \( 1 + 1.41iT - 5T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
23 \( 1 - 1.41iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 + 4.24iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 - 1.41iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 - 4.24iT - 79T^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 8.48iT - 97T^{2} \)
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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

−10.28509335105820461948263180658, −9.300069343039410276111258811739, −8.485436469931972758114408459179, −7.975050154688608856900829809602, −7.05096874502569349896050384171, −5.36109871006620078070576367038, −4.82959454165294789558560807376, −4.19956131266350948543110261800, −2.70008758134688100532996270514, −1.55648271257340826115003204993, 1.17183039492744993515610633033, 2.34271117101274267172122724383, 3.38603598172315336102113580230, 4.57640624267292969372136466235, 5.84669845906003147891072410212, 6.64270932641587577872259995017, 7.51681246314878098211096912736, 8.444961022619276800782710193718, 8.668407020429631791837187438692, 9.978671065961800473017045413671

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