↑

Properties

Label 2-2960-740.599-c0-0-1
Degree $2$
Conductor $2960$
Sign $0.801 - 0.597i$
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)5-s + (−0.766 − 0.642i)9-s + (1.11 + 1.32i)13-s + (1.26 − 1.50i)17-s + (−0.499 + 0.866i)25-s + (−0.766 + 1.32i)29-s + (0.766 − 0.642i)37-s + (−0.266 + 0.223i)41-s + (0.173 − 0.984i)45-s + (0.939 − 0.342i)49-s + (1.70 + 0.300i)53-s + (−1.17 + 0.984i)61-s + (−0.592 + 1.62i)65-s + 1.73i·73-s + (0.173 + 0.984i)81-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s + (−0.766 − 0.642i)9-s + (1.11 + 1.32i)13-s + (1.26 − 1.50i)17-s + (−0.499 + 0.866i)25-s + (−0.766 + 1.32i)29-s + (0.766 − 0.642i)37-s + (−0.266 + 0.223i)41-s + (0.173 − 0.984i)45-s + (0.939 − 0.342i)49-s + (1.70 + 0.300i)53-s + (−1.17 + 0.984i)61-s + (−0.592 + 1.62i)65-s + 1.73i·73-s + (0.173 + 0.984i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.801 - 0.597i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (2079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ 0.801 - 0.597i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.376855214\)
\(L(\frac12)\) \(\approx\) \(1.376855214\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.766 + 0.642i)T \)
good3 \( 1 + (0.766 + 0.642i)T^{2} \)
7 \( 1 + (-0.939 + 0.342i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.11 - 1.32i)T + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (-1.26 + 1.50i)T + (-0.173 - 0.984i)T^{2} \)
19 \( 1 + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-1.70 - 0.300i)T + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.173 + 0.984i)T^{2} \)
89 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (1.11 - 0.642i)T + (0.5 - 0.866i)T^{2} \)
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

−9.200961441902208500654713665784, −8.363149074606064040824933703168, −7.17895287739254771349860962480, −6.91041022316860394555703937762, −5.84595626631591809396982505678, −5.49574409554087649665100933948, −4.12408902638415485054756385436, −3.31963559672060161439984066105, −2.57679909886225275027461410748, −1.30847329791483175148100414287, 1.01686930626138233168466744919, 2.11467765686802138023919121503, 3.26071587161163226157379346029, 4.10326813641609086752666829977, 5.22812130378557127265612344078, 5.80952190665740276653264324464, 6.17024902313486835387786951113, 7.76708683159845094897864442118, 8.089805848470896476771890316538, 8.689400831560563865216701445496

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