↑

Properties

Label 2-2960-2960.1173-c0-0-0
Degree $2$
Conductor $2960$
Sign $0.888 - 0.459i$
Analytic cond. $1.47723$
Root an. cond. $1.21541$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·6-s + (1.36 − 0.366i)7-s − 0.999·8-s + 0.999·10-s + (1 + i)11-s + (0.499 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (1 + 0.999i)14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.499 + 0.866i)20-s + (0.366 − 1.36i)21-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·6-s + (1.36 − 0.366i)7-s − 0.999·8-s + 0.999·10-s + (1 + i)11-s + (0.499 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (1 + 0.999i)14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.499 + 0.866i)20-s + (0.366 − 1.36i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.888 - 0.459i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (1173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ 0.888 - 0.459i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.266765704\)
\(L(\frac12)\) \(\approx\) \(2.266765704\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-1 - i)T + iT^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (1 - i)T - iT^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + T + T^{2} \)
41 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - iT^{2} \)
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

−8.744918097350213145352742707939, −8.061436236222512281588085301376, −7.39543012121670606473748436837, −6.96433300783165662911454341348, −5.99764561758293062944112577559, −4.94522656567518875200642168861, −4.62262723815246689999144778096, −3.70871764324636829082618824467, −1.92000757654363099373870494719, −1.70567787575489636158966377567, 1.43355795155110563840772398439, 2.48009492137445722880490452763, 3.24186271827288508707532292692, 4.01062486430068073345701601785, 4.77196016251556714169107841172, 5.68385665024294844466609732940, 6.23657639029656423688377238112, 7.43340238240120863292042633022, 8.548031844796485534670684792423, 8.929135413885952907577493325162

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