↑

Properties

Label 2-2960-740.219-c0-0-1
Degree $2$
Conductor $2960$
Sign $0.997 - 0.0659i$
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  + (0.939 − 0.342i)5-s + (−0.173 + 0.984i)9-s + (1.70 − 0.300i)13-s + (−0.673 − 0.118i)17-s + (0.766 − 0.642i)25-s + (−0.173 + 0.300i)29-s + (−0.173 − 0.984i)37-s + (0.326 + 1.85i)41-s + (0.173 + 0.984i)45-s + (−0.766 − 0.642i)49-s + (0.592 + 1.62i)53-s + (−0.0603 − 0.342i)61-s + (1.5 − 0.866i)65-s − 1.73i·73-s + (−0.939 − 0.342i)81-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)5-s + (−0.173 + 0.984i)9-s + (1.70 − 0.300i)13-s + (−0.673 − 0.118i)17-s + (0.766 − 0.642i)25-s + (−0.173 + 0.300i)29-s + (−0.173 − 0.984i)37-s + (0.326 + 1.85i)41-s + (0.173 + 0.984i)45-s + (−0.766 − 0.642i)49-s + (0.592 + 1.62i)53-s + (−0.0603 − 0.342i)61-s + (1.5 − 0.866i)65-s − 1.73i·73-s + (−0.939 − 0.342i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.997 - 0.0659i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ 0.997 - 0.0659i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.548187480\)
\(L(\frac12)\) \(\approx\) \(1.548187480\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (0.173 + 0.984i)T \)
good3 \( 1 + (0.173 - 0.984i)T^{2} \)
7 \( 1 + (0.766 + 0.642i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.70 + 0.300i)T + (0.939 - 0.342i)T^{2} \)
17 \( 1 + (0.673 + 0.118i)T + (0.939 + 0.342i)T^{2} \)
19 \( 1 + (-0.173 + 0.984i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.939 - 0.342i)T^{2} \)
89 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (1.70 - 0.984i)T + (0.5 - 0.866i)T^{2} \)
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

−8.884638442422024844836210389299, −8.326583930935723024035476229852, −7.52793614719633083755645929917, −6.43532673747273445571075348313, −5.94051524257632335582977654164, −5.13920711989103652357700203723, −4.36994313719480618681011403862, −3.23705085395433695301731454958, −2.22019411561921900914954899334, −1.30613948233597428249285305261, 1.21060084276981517312490505339, 2.26268982932807197356148985107, 3.37459283632182272276564011862, 4.02643698068394531225910843631, 5.23353273421880285309750787769, 6.09242862351051351904097263499, 6.45310651844756990536185970874, 7.21096805929649188470803835735, 8.502992229712512331656653344728, 8.830162358272011834244593252361

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