↑

Properties

Label 2-520-1.1-c1-0-0
Degree 22
Conductor 520520
Sign 11
Analytic cond. 4.152224.15222
Root an. cond. 2.037692.03769
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.73·3-s − 5-s − 3.46·7-s + 4.46·9-s − 0.732·11-s + 13-s + 2.73·15-s + 0.535·17-s + 0.732·19-s + 9.46·21-s + 2.73·23-s + 25-s − 3.99·27-s + 2.53·29-s + 10.1·31-s + 2·33-s + 3.46·35-s − 2.73·39-s + 7.46·41-s − 10.7·43-s − 4.46·45-s + 11.4·47-s + 4.99·49-s − 1.46·51-s − 7.46·53-s + 0.732·55-s − 2·57-s + ⋯
L(s)  = 1  − 1.57·3-s − 0.447·5-s − 1.30·7-s + 1.48·9-s − 0.220·11-s + 0.277·13-s + 0.705·15-s + 0.129·17-s + 0.167·19-s + 2.06·21-s + 0.569·23-s + 0.200·25-s − 0.769·27-s + 0.470·29-s + 1.83·31-s + 0.348·33-s + 0.585·35-s − 0.437·39-s + 1.16·41-s − 1.63·43-s − 0.665·45-s + 1.67·47-s + 0.714·49-s − 0.205·51-s − 1.02·53-s + 0.0987·55-s − 0.264·57-s + ⋯

Functional equation

Λ(s)=(520s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(520s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 520520    =    23⋅5⋅132^{3} \cdot 5 \cdot 13
Sign: 11
Analytic conductor: 4.152224.15222
Root analytic conductor: 2.037692.03769
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 520, ( :1/2), 1)(2,\ 520,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.56780709940.5678070994
L(12)L(\frac12) ≈\approx 0.56780709940.5678070994
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1+T 1 + T
13 1−T 1 - T
good3 1+2.73T+3T2 1 + 2.73T + 3T^{2}
7 1+3.46T+7T2 1 + 3.46T + 7T^{2}
11 1+0.732T+11T2 1 + 0.732T + 11T^{2}
17 1−0.535T+17T2 1 - 0.535T + 17T^{2}
19 1−0.732T+19T2 1 - 0.732T + 19T^{2}
23 1−2.73T+23T2 1 - 2.73T + 23T^{2}
29 1−2.53T+29T2 1 - 2.53T + 29T^{2}
31 1−10.1T+31T2 1 - 10.1T + 31T^{2}
37 1+37T2 1 + 37T^{2}
41 1−7.46T+41T2 1 - 7.46T + 41T^{2}
43 1+10.7T+43T2 1 + 10.7T + 43T^{2}
47 1−11.4T+47T2 1 - 11.4T + 47T^{2}
53 1+7.46T+53T2 1 + 7.46T + 53T^{2}
59 1+11.6T+59T2 1 + 11.6T + 59T^{2}
61 1−8.39T+61T2 1 - 8.39T + 61T^{2}
67 1−0.928T+67T2 1 - 0.928T + 67T^{2}
71 1−12.7T+71T2 1 - 12.7T + 71T^{2}
73 1−10.9T+73T2 1 - 10.9T + 73T^{2}
79 1+13.4T+79T2 1 + 13.4T + 79T^{2}
83 1−10.3T+83T2 1 - 10.3T + 83T^{2}
89 1−0.928T+89T2 1 - 0.928T + 89T^{2}
97 1+11.8T+97T2 1 + 11.8T + 97T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−10.90408677099640618740331508210, −10.20414253020792576095761356254, −9.368171480062089548096755888861, −8.060210896072692909797751119787, −6.83133973832531209909933199725, −6.34776039853551283174323004610, −5.37276639251637763627303023232, −4.34215244154987436195976911636, −3.04747144414217232239792058212, −0.72301461500310057444827065062, 0.72301461500310057444827065062, 3.04747144414217232239792058212, 4.34215244154987436195976911636, 5.37276639251637763627303023232, 6.34776039853551283174323004610, 6.83133973832531209909933199725, 8.060210896072692909797751119787, 9.368171480062089548096755888861, 10.20414253020792576095761356254, 10.90408677099640618740331508210

Graph of the ZZ-function along the critical line