Properties

Label 2-960-12.11-c3-0-0
Degree $2$
Conductor $960$
Sign $-0.926 - 0.376i$
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  + (1.95 − 4.81i)3-s + 5i·5-s + 20.3i·7-s + (−19.3 − 18.8i)9-s + 15.2·11-s − 27.7·13-s + (24.0 + 9.77i)15-s + 92.5i·17-s − 127. i·19-s + (98.0 + 39.8i)21-s + 51.1·23-s − 25·25-s + (−128. + 56.3i)27-s − 99.2i·29-s + 25.8i·31-s + ⋯
L(s)  = 1  + (0.376 − 0.926i)3-s + 0.447i·5-s + 1.09i·7-s + (−0.716 − 0.697i)9-s + 0.419·11-s − 0.591·13-s + (0.414 + 0.168i)15-s + 1.31i·17-s − 1.54i·19-s + (1.01 + 0.413i)21-s + 0.463·23-s − 0.200·25-s + (−0.915 + 0.401i)27-s − 0.635i·29-s + 0.149i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.376i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.926 - 0.376i$
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: \((2,\ 960,\ (\ :3/2),\ -0.926 - 0.376i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1051659160\)
\(L(\frac12)\) \(\approx\) \(0.1051659160\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.95 + 4.81i)T \)
5 \( 1 - 5iT \)
good7 \( 1 - 20.3iT - 343T^{2} \)
11 \( 1 - 15.2T + 1.33e3T^{2} \)
13 \( 1 + 27.7T + 2.19e3T^{2} \)
17 \( 1 - 92.5iT - 4.91e3T^{2} \)
19 \( 1 + 127. iT - 6.85e3T^{2} \)
23 \( 1 - 51.1T + 1.21e4T^{2} \)
29 \( 1 + 99.2iT - 2.43e4T^{2} \)
31 \( 1 - 25.8iT - 2.97e4T^{2} \)
37 \( 1 + 356.T + 5.06e4T^{2} \)
41 \( 1 + 292. iT - 6.89e4T^{2} \)
43 \( 1 - 521. iT - 7.95e4T^{2} \)
47 \( 1 + 573.T + 1.03e5T^{2} \)
53 \( 1 + 305. iT - 1.48e5T^{2} \)
59 \( 1 + 295.T + 2.05e5T^{2} \)
61 \( 1 + 326.T + 2.26e5T^{2} \)
67 \( 1 + 299. iT - 3.00e5T^{2} \)
71 \( 1 + 653.T + 3.57e5T^{2} \)
73 \( 1 - 504.T + 3.89e5T^{2} \)
79 \( 1 + 110. iT - 4.93e5T^{2} \)
83 \( 1 + 1.47e3T + 5.71e5T^{2} \)
89 \( 1 + 772. iT - 7.04e5T^{2} \)
97 \( 1 + 26.1T + 9.12e5T^{2} \)
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

−9.834941356007534520887399524720, −8.960755933816433738632011685622, −8.433547118453732263662505896876, −7.42856898167080592230913232517, −6.62565940044158337399702722507, −5.97038102279118086448742840998, −4.86029141691017624835652815767, −3.39251035690016192900907715580, −2.53083077727456322366008614527, −1.61827679633992664991940625838, 0.02362975562704316607174619754, 1.51839245666203305436447804590, 3.05015850775215286093019723529, 3.91633687469922988168743649699, 4.74823920291499836983868449334, 5.51402294747150626335802456186, 6.88573354744927603317952532338, 7.66558171699736793970366882020, 8.563209522060122935851808738794, 9.397870055736578215492818795726

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