↑

Properties

Label 2-960-12.11-c1-0-12
Degree $2$
Conductor $960$
Sign $0.489 - 0.871i$
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  + (1.51 + 0.848i)3-s − i·5-s + 3.02i·7-s + (1.56 + 2.56i)9-s − 1.32·11-s + 5.12·13-s + (0.848 − 1.51i)15-s − 2i·17-s + 1.32i·19-s + (−2.56 + 4.56i)21-s + 0.371·23-s − 25-s + (0.185 + 5.19i)27-s + 3.12i·29-s + 4.71i·31-s + ⋯
L(s)  = 1  + (0.871 + 0.489i)3-s − 0.447i·5-s + 1.14i·7-s + (0.520 + 0.853i)9-s − 0.399·11-s + 1.42·13-s + (0.218 − 0.389i)15-s − 0.485i·17-s + 0.303i·19-s + (−0.558 + 0.995i)21-s + 0.0775·23-s − 0.200·25-s + (0.0357 + 0.999i)27-s + 0.579i·29-s + 0.847i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.489 - 0.871i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.489 - 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88974 + 1.10612i\)
\(L(\frac12)\) \(\approx\) \(1.88974 + 1.10612i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.51 - 0.848i)T \)
5 \( 1 + iT \)
good7 \( 1 - 3.02iT - 7T^{2} \)
11 \( 1 + 1.32T + 11T^{2} \)
13 \( 1 - 5.12T + 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 1.32iT - 19T^{2} \)
23 \( 1 - 0.371T + 23T^{2} \)
29 \( 1 - 3.12iT - 29T^{2} \)
31 \( 1 - 4.71iT - 31T^{2} \)
37 \( 1 + 5.12T + 37T^{2} \)
41 \( 1 - 1.12iT - 41T^{2} \)
43 \( 1 - 7.73iT - 43T^{2} \)
47 \( 1 - 3.02T + 47T^{2} \)
53 \( 1 + 12.2iT - 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 + 3.12T + 61T^{2} \)
67 \( 1 + 4.34iT - 67T^{2} \)
71 \( 1 - 3.39T + 71T^{2} \)
73 \( 1 - 8.24T + 73T^{2} \)
79 \( 1 + 8.10iT - 79T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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

−10.00947850772251084971778040230, −9.132780614056953960585258794139, −8.597005172014709919104332967020, −8.047695209833354272128608989606, −6.81311806579418750232078707112, −5.63881629234477498565990944576, −4.94416816240168934404515667837, −3.75024211311188530138845145294, −2.85038785527138762740415210921, −1.65522069524231165117366213529, 1.01909066393524675857853061653, 2.36431209826502088299901049207, 3.60362251330781105921286427909, 4.12326062997423796001799214377, 5.76328478490022439757087503540, 6.71961595879231862913097590575, 7.37030832509953015702063078176, 8.180758634309556069049138644552, 8.887483911903095481013459423093, 9.952385589514246451193770236643

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