↑

Properties

Label 2-960-40.29-c1-0-9
Degree $2$
Conductor $960$
Sign $0.316 - 0.948i$
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  + 3-s + (1 + 2i)5-s + 2i·7-s + 9-s + 2i·11-s + 2·13-s + (1 + 2i)15-s − 4i·17-s − 2i·19-s + 2i·21-s + 6i·23-s + (−3 + 4i)25-s + 27-s + 4i·29-s − 8·31-s + ⋯
L(s)  = 1  + 0.577·3-s + (0.447 + 0.894i)5-s + 0.755i·7-s + 0.333·9-s + 0.603i·11-s + 0.554·13-s + (0.258 + 0.516i)15-s − 0.970i·17-s − 0.458i·19-s + 0.436i·21-s + 1.25i·23-s + (−0.600 + 0.800i)25-s + 0.192·27-s + 0.742i·29-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.316 - 0.948i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67343 + 1.20614i\)
\(L(\frac12)\) \(\approx\) \(1.67343 + 1.20614i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 + (-1 - 2i)T \)
good7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 - 10T + 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + 8iT - 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 16iT - 97T^{2} \)
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

−9.945390098040302828120174546708, −9.432003937323121926259927836612, −8.678017027444621528053369816238, −7.52506845560616517194788636232, −6.96503066497852641375843745291, −5.90884910800367383644708730339, −5.03893009182737536907298803387, −3.64827733511076420208503472582, −2.76749875908977418541463166365, −1.80107749269515286491746855798, 0.946090791114472922337670514859, 2.18419089687037272141547525349, 3.68196767008157501868807950936, 4.32809844973046431951389949443, 5.57745285432679839474960107288, 6.36361920797486524263920111276, 7.49886976770475233595022465125, 8.464382178538791126905696001078, 8.758514178063275267563581222078, 9.927552420453697507127993756516

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