↑

Properties

Label 2-960-60.59-c1-0-36
Degree $2$
Conductor $960$
Sign $-0.267 + 0.963i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − i)3-s + (−1.73 + 1.41i)5-s + (1.00 − 2.82i)9-s − 4.89·11-s − 4.89i·13-s + (−1.03 + 3.73i)15-s + 3.46·17-s − 3.46i·19-s − 6i·23-s + (0.999 − 4.89i)25-s + (−1.41 − 5.00i)27-s − 2.82i·29-s − 3.46i·31-s + (−6.92 + 4.89i)33-s + 4.89i·37-s + ⋯
L(s)  = 1  + (0.816 − 0.577i)3-s + (−0.774 + 0.632i)5-s + (0.333 − 0.942i)9-s − 1.47·11-s − 1.35i·13-s + (−0.267 + 0.963i)15-s + 0.840·17-s − 0.794i·19-s − 1.25i·23-s + (0.199 − 0.979i)25-s + (−0.272 − 0.962i)27-s − 0.525i·29-s − 0.622i·31-s + (−1.20 + 0.852i)33-s + 0.805i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.267 + 0.963i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.267 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.821919 - 1.08095i\)
\(L(\frac12)\) \(\approx\) \(0.821919 - 1.08095i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + i)T \)
5 \( 1 + (1.73 - 1.41i)T \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 + 4.89iT - 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 - 4.89iT - 37T^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 - 8.48T + 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + 4.89T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 8.48T + 67T^{2} \)
71 \( 1 + 9.79T + 71T^{2} \)
73 \( 1 - 9.79iT - 73T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + 5.65iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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

−9.960935036395007266417638425201, −8.622892549162665248408976660998, −7.940950905267887008578172836479, −7.56872814189466738156830095771, −6.57976001880268308031826268409, −5.49415004956789185798081543722, −4.25603133905767588766897207109, −2.99065885592311772998812773814, −2.63088552283868851373844411816, −0.56447623842158636909037049944, 1.72895977703801034567059020261, 3.11158212028217115652841986065, 3.98076265085130387070085898363, 4.86638835357332674377193743828, 5.68784268034844097247518107578, 7.46915234965726146819716876723, 7.64761827838493410703353932326, 8.720812047839733088248734318947, 9.264881173433288849278811985389, 10.21375156375044352492763195471

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