↑

Properties

Label 2-960-12.11-c3-0-68
Degree $2$
Conductor $960$
Sign $-0.868 + 0.496i$
Analytic cond. $56.6418$
Root an. cond. $7.52607$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.57 − 4.51i)3-s − 5i·5-s + 10.9i·7-s + (−13.7 + 23.2i)9-s − 7.15·11-s + 38.2·13-s + (−22.5 + 12.8i)15-s + 40.5i·17-s + 25.4i·19-s + (49.5 − 28.3i)21-s − 17.1·23-s − 25·25-s + (140. + 1.84i)27-s − 250. i·29-s − 219. i·31-s + ⋯
L(s)  = 1  + (−0.496 − 0.868i)3-s − 0.447i·5-s + 0.593i·7-s + (−0.507 + 0.861i)9-s − 0.196·11-s + 0.816·13-s + (−0.388 + 0.221i)15-s + 0.578i·17-s + 0.307i·19-s + (0.515 − 0.294i)21-s − 0.155·23-s − 0.200·25-s + (0.999 + 0.0131i)27-s − 1.60i·29-s − 1.26i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 + 0.496i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.868 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.868 + 0.496i$
Analytic conductor: \(56.6418\)
Root analytic conductor: \(7.52607\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :3/2),\ -0.868 + 0.496i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9274922132\)
\(L(\frac12)\) \(\approx\) \(0.9274922132\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.57 + 4.51i)T \)
5 \( 1 + 5iT \)
good7 \( 1 - 10.9iT - 343T^{2} \)
11 \( 1 + 7.15T + 1.33e3T^{2} \)
13 \( 1 - 38.2T + 2.19e3T^{2} \)
17 \( 1 - 40.5iT - 4.91e3T^{2} \)
19 \( 1 - 25.4iT - 6.85e3T^{2} \)
23 \( 1 + 17.1T + 1.21e4T^{2} \)
29 \( 1 + 250. iT - 2.43e4T^{2} \)
31 \( 1 + 219. iT - 2.97e4T^{2} \)
37 \( 1 - 139.T + 5.06e4T^{2} \)
41 \( 1 - 148. iT - 6.89e4T^{2} \)
43 \( 1 - 84.1iT - 7.95e4T^{2} \)
47 \( 1 + 429.T + 1.03e5T^{2} \)
53 \( 1 + 559. iT - 1.48e5T^{2} \)
59 \( 1 + 771.T + 2.05e5T^{2} \)
61 \( 1 - 557.T + 2.26e5T^{2} \)
67 \( 1 + 804. iT - 3.00e5T^{2} \)
71 \( 1 - 643.T + 3.57e5T^{2} \)
73 \( 1 + 89.5T + 3.89e5T^{2} \)
79 \( 1 - 100. iT - 4.93e5T^{2} \)
83 \( 1 + 1.19e3T + 5.71e5T^{2} \)
89 \( 1 - 251. iT - 7.04e5T^{2} \)
97 \( 1 - 1.64e3T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.227828491482131540689451566048, −8.107093064753068824709856400493, −7.931479320155822892378069644111, −6.46157688370578544047723209354, −6.00695111774774890858060645671, −5.11514012902195592031578362094, −3.95612924326658177978627809694, −2.50570960947614952953022738368, −1.51958948460525832487959842523, −0.28127324345017943515533295054, 1.12738335229240899454247460519, 2.92312760976941331229380992045, 3.74422240475530507985430536084, 4.72206705044156293375697712618, 5.58804131247059678908614418443, 6.57661510362298705189455964375, 7.28930818369242569726548938595, 8.520170096717592669053683011837, 9.238458281982778299672740608588, 10.20324772211634909042605349675

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