↑

Properties

Label 2-720-80.27-c1-0-7
Degree $2$
Conductor $720$
Sign $0.596 - 0.802i$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
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  + (−0.567 − 1.29i)2-s + (−1.35 + 1.47i)4-s + (1.42 + 1.72i)5-s + (−1.60 − 1.60i)7-s + (2.67 + 0.920i)8-s + (1.42 − 2.82i)10-s + (−0.754 + 0.754i)11-s + 5.94i·13-s + (−1.16 + 2.98i)14-s + (−0.327 − 3.98i)16-s + (−1.95 − 1.95i)17-s + (0.780 − 0.780i)19-s + (−4.46 − 0.247i)20-s + (1.40 + 0.548i)22-s + (−4.93 + 4.93i)23-s + ⋯
L(s)  = 1  + (−0.401 − 0.915i)2-s + (−0.677 + 0.735i)4-s + (0.635 + 0.771i)5-s + (−0.605 − 0.605i)7-s + (0.945 + 0.325i)8-s + (0.451 − 0.892i)10-s + (−0.227 + 0.227i)11-s + 1.64i·13-s + (−0.311 + 0.797i)14-s + (−0.0817 − 0.996i)16-s + (−0.474 − 0.474i)17-s + (0.179 − 0.179i)19-s + (−0.998 − 0.0553i)20-s + (0.299 + 0.117i)22-s + (−1.02 + 1.02i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.596 - 0.802i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.596 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.776579 + 0.390504i\)
\(L(\frac12)\) \(\approx\) \(0.776579 + 0.390504i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.567 + 1.29i)T \)
3 \( 1 \)
5 \( 1 + (-1.42 - 1.72i)T \)
good7 \( 1 + (1.60 + 1.60i)T + 7iT^{2} \)
11 \( 1 + (0.754 - 0.754i)T - 11iT^{2} \)
13 \( 1 - 5.94iT - 13T^{2} \)
17 \( 1 + (1.95 + 1.95i)T + 17iT^{2} \)
19 \( 1 + (-0.780 + 0.780i)T - 19iT^{2} \)
23 \( 1 + (4.93 - 4.93i)T - 23iT^{2} \)
29 \( 1 + (-1.44 - 1.44i)T + 29iT^{2} \)
31 \( 1 + 3.60iT - 31T^{2} \)
37 \( 1 - 10.2iT - 37T^{2} \)
41 \( 1 - 6.93iT - 41T^{2} \)
43 \( 1 - 9.91iT - 43T^{2} \)
47 \( 1 + (0.104 - 0.104i)T - 47iT^{2} \)
53 \( 1 - 4.03T + 53T^{2} \)
59 \( 1 + (-3.46 - 3.46i)T + 59iT^{2} \)
61 \( 1 + (-0.680 + 0.680i)T - 61iT^{2} \)
67 \( 1 + 9.04iT - 67T^{2} \)
71 \( 1 - 3.64T + 71T^{2} \)
73 \( 1 + (2.94 + 2.94i)T + 73iT^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 - 4.23T + 83T^{2} \)
89 \( 1 + 0.0426T + 89T^{2} \)
97 \( 1 + (1.91 + 1.91i)T + 97iT^{2} \)
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

−10.42522901375622447675547073415, −9.653076828011526157910794330898, −9.362407600193630459424799681562, −8.016671889169324981829326890929, −7.05018101577061460694741408728, −6.34785480970148642272104823014, −4.81361721175941260352907859087, −3.80436607549240358090492489571, −2.74599844725741034439552346995, −1.64582762771007513126847414122, 0.51443373386913311930646162525, 2.30001520835118670424065807324, 3.98205606924535507180364140781, 5.37012296133602617458364668968, 5.70825795549883032962888324768, 6.63910983576016135927767340040, 7.87839274753931253695862020492, 8.568404690250136231526813038746, 9.189763013138128386126382604279, 10.21353056714258961563219592314

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