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Properties

Label 2-960-192.131-c1-0-93
Degree $2$
Conductor $960$
Sign $0.158 + 0.987i$
Analytic cond. $7.66563$
Root an. cond. $2.76868$
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  + (−1.31 + 0.523i)2-s + (0.243 + 1.71i)3-s + (1.45 − 1.37i)4-s + (0.555 − 0.831i)5-s + (−1.21 − 2.12i)6-s + (−0.799 − 1.92i)7-s + (−1.18 + 2.56i)8-s + (−2.88 + 0.834i)9-s + (−0.294 + 1.38i)10-s + (−2.67 + 0.531i)11-s + (2.71 + 2.15i)12-s + (−4.23 + 2.82i)13-s + (2.06 + 2.11i)14-s + (1.56 + 0.750i)15-s + (0.214 − 3.99i)16-s + (0.635 − 0.635i)17-s + ⋯
L(s)  = 1  + (−0.928 + 0.370i)2-s + (0.140 + 0.990i)3-s + (0.725 − 0.687i)4-s + (0.248 − 0.371i)5-s + (−0.497 − 0.867i)6-s + (−0.302 − 0.729i)7-s + (−0.419 + 0.907i)8-s + (−0.960 + 0.278i)9-s + (−0.0931 + 0.437i)10-s + (−0.806 + 0.160i)11-s + (0.783 + 0.621i)12-s + (−1.17 + 0.784i)13-s + (0.550 + 0.565i)14-s + (0.403 + 0.193i)15-s + (0.0535 − 0.998i)16-s + (0.154 − 0.154i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.158 + 0.987i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.158 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.305895 - 0.260819i\)
\(L(\frac12)\) \(\approx\) \(0.305895 - 0.260819i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 - 0.523i)T \)
3 \( 1 + (-0.243 - 1.71i)T \)
5 \( 1 + (-0.555 + 0.831i)T \)
good7 \( 1 + (0.799 + 1.92i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (2.67 - 0.531i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (4.23 - 2.82i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (-0.635 + 0.635i)T - 17iT^{2} \)
19 \( 1 + (-6.79 + 4.54i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (0.709 - 1.71i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.845 + 4.24i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + 2.99T + 31T^{2} \)
37 \( 1 + (-1.81 + 2.71i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (1.29 + 0.538i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (11.2 - 2.23i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (7.49 + 7.49i)T + 47iT^{2} \)
53 \( 1 + (2.51 + 12.6i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (7.64 + 5.10i)T + (22.5 + 54.5i)T^{2} \)
61 \( 1 + (-1.87 + 9.40i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (-10.5 - 2.09i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (8.81 - 3.65i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-10.5 - 4.36i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (9.58 + 9.58i)T + 79iT^{2} \)
83 \( 1 + (3.05 + 4.56i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (-1.06 + 0.441i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 - 10.3iT - 97T^{2} \)
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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

−9.806936526070670843207424405488, −9.295567323099141821412962945097, −8.241095375365816005757882814524, −7.45335784236099911876475300947, −6.64156582142519861409485742874, −5.25828082531179402573775170310, −4.89844650240272075707175555651, −3.36439827268069885655383997627, −2.17114022173258904558423019218, −0.23826365576447382681356265724, 1.48115690693453090691423653912, 2.74517760889513790880995044254, 3.13123236107740018684642660648, 5.34201686048145278247029747816, 6.09626012348096084756831406173, 7.16821827512033498298904287433, 7.75342265778438322276439984496, 8.424146747455360504015214469234, 9.461287402266497184938428006074, 10.07220114824065347725212644724

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