↑

Properties

Label 2-960-24.11-c1-0-2
Degree $2$
Conductor $960$
Sign $0.0267 - 0.999i$
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  + (0.492 − 1.66i)3-s − 5-s + 1.32i·7-s + (−2.51 − 1.63i)9-s + 5.02i·11-s + 6.73i·13-s + (−0.492 + 1.66i)15-s − 5.43i·17-s − 6.73·19-s + (2.19 + 0.651i)21-s − 5.75·23-s + 25-s + (−3.95 + 3.36i)27-s − 1.61·29-s + 7.02i·31-s + ⋯
L(s)  = 1  + (0.284 − 0.958i)3-s − 0.447·5-s + 0.499i·7-s + (−0.838 − 0.545i)9-s + 1.51i·11-s + 1.86i·13-s + (−0.127 + 0.428i)15-s − 1.31i·17-s − 1.54·19-s + (0.478 + 0.142i)21-s − 1.19·23-s + 0.200·25-s + (−0.761 + 0.648i)27-s − 0.299·29-s + 1.26i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0267 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0267 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.0267 - 0.999i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.0267 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.597128 + 0.581362i\)
\(L(\frac12)\) \(\approx\) \(0.597128 + 0.581362i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.492 + 1.66i)T \)
5 \( 1 + T \)
good7 \( 1 - 1.32iT - 7T^{2} \)
11 \( 1 - 5.02iT - 11T^{2} \)
13 \( 1 - 6.73iT - 13T^{2} \)
17 \( 1 + 5.43iT - 17T^{2} \)
19 \( 1 + 6.73T + 19T^{2} \)
23 \( 1 + 5.75T + 23T^{2} \)
29 \( 1 + 1.61T + 29T^{2} \)
31 \( 1 - 7.02iT - 31T^{2} \)
37 \( 1 - 4.76iT - 37T^{2} \)
41 \( 1 - 3.27iT - 41T^{2} \)
43 \( 1 - 2.95T + 43T^{2} \)
47 \( 1 - 0.795T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 + 1.61iT - 59T^{2} \)
61 \( 1 + 5.24iT - 61T^{2} \)
67 \( 1 + 5.56T + 67T^{2} \)
71 \( 1 + 4.95T + 71T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
79 \( 1 + 6.44iT - 79T^{2} \)
83 \( 1 - 8.67iT - 83T^{2} \)
89 \( 1 + 16.4iT - 89T^{2} \)
97 \( 1 - 8.64T + 97T^{2} \)
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

−10.10087707180344808044751411633, −9.158188203539845076309086826451, −8.641999577792661643714757698662, −7.54138705771961457923649202118, −6.93416813644304600810588457004, −6.27747873202660861569051735168, −4.86672930371845481441394534579, −4.03607916747701819084033727627, −2.49241315033723664306535747228, −1.77514820813019085188139069986, 0.35419094117649581559268952278, 2.54994475611538998064779711273, 3.74738335313974300527719470341, 4.08757034199464053081805774402, 5.64149964111957311361365094319, 6.00203525510236094913704264958, 7.65306641552289297584277690349, 8.297076796541568920711047924912, 8.770143404028469770968350094709, 10.08176735054384969324889582318

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