↑

Properties

Label 2-520-1.1-c1-0-3
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  + 0.732·3-s − 5-s + 3.46·7-s − 2.46·9-s + 2.73·11-s + 13-s − 0.732·15-s + 7.46·17-s − 2.73·19-s + 2.53·21-s − 0.732·23-s + 25-s − 4·27-s + 9.46·29-s − 0.196·31-s + 2·33-s − 3.46·35-s + 0.732·39-s + 0.535·41-s − 7.26·43-s + 2.46·45-s + 4.53·47-s + 4.99·49-s + 5.46·51-s − 0.535·53-s − 2.73·55-s − 2·57-s + ⋯
L(s)  = 1  + 0.422·3-s − 0.447·5-s + 1.30·7-s − 0.821·9-s + 0.823·11-s + 0.277·13-s − 0.189·15-s + 1.81·17-s − 0.626·19-s + 0.553·21-s − 0.152·23-s + 0.200·25-s − 0.769·27-s + 1.75·29-s − 0.0352·31-s + 0.348·33-s − 0.585·35-s + 0.117·39-s + 0.0836·41-s − 1.10·43-s + 0.367·45-s + 0.661·47-s + 0.714·49-s + 0.765·51-s − 0.0736·53-s − 0.368·55-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 1.7630024301.763002430
L(12)L(\frac12) ≈\approx 1.7630024301.763002430
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 1−0.732T+3T2 1 - 0.732T + 3T^{2}
7 1−3.46T+7T2 1 - 3.46T + 7T^{2}
11 1−2.73T+11T2 1 - 2.73T + 11T^{2}
17 1−7.46T+17T2 1 - 7.46T + 17T^{2}
19 1+2.73T+19T2 1 + 2.73T + 19T^{2}
23 1+0.732T+23T2 1 + 0.732T + 23T^{2}
29 1−9.46T+29T2 1 - 9.46T + 29T^{2}
31 1+0.196T+31T2 1 + 0.196T + 31T^{2}
37 1+37T2 1 + 37T^{2}
41 1−0.535T+41T2 1 - 0.535T + 41T^{2}
43 1+7.26T+43T2 1 + 7.26T + 43T^{2}
47 1−4.53T+47T2 1 - 4.53T + 47T^{2}
53 1+0.535T+53T2 1 + 0.535T + 53T^{2}
59 1−5.66T+59T2 1 - 5.66T + 59T^{2}
61 1+12.3T+61T2 1 + 12.3T + 61T^{2}
67 1+12.9T+67T2 1 + 12.9T + 67T^{2}
71 1−9.26T+71T2 1 - 9.26T + 71T^{2}
73 1+2.92T+73T2 1 + 2.92T + 73T^{2}
79 1+6.53T+79T2 1 + 6.53T + 79T^{2}
83 1+10.3T+83T2 1 + 10.3T + 83T^{2}
89 1+12.9T+89T2 1 + 12.9T + 89T^{2}
97 1−15.8T+97T2 1 - 15.8T + 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

−10.97600575780112934316907505504, −10.02924581371377934703209868279, −8.793363143840068806807251551510, −8.274658006708178020450122064181, −7.53844637418609047564072990656, −6.23744849259013932365528684669, −5.16920999808620702659828679307, −4.09043065945765163757391542059, −2.95818306668519194005910648112, −1.38230213712586315015274173446, 1.38230213712586315015274173446, 2.95818306668519194005910648112, 4.09043065945765163757391542059, 5.16920999808620702659828679307, 6.23744849259013932365528684669, 7.53844637418609047564072990656, 8.274658006708178020450122064181, 8.793363143840068806807251551510, 10.02924581371377934703209868279, 10.97600575780112934316907505504

Graph of the ZZ-function along the critical line