↑

Properties

Label 2-520-1.1-c1-0-4
Degree 22
Conductor 520520
Sign 11
Analytic cond. 4.152224.15222
Root an. cond. 2.037692.03769
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 5-s + 9-s + 2·11-s + 13-s + 2·15-s + 2·17-s + 2·19-s + 2·23-s + 25-s − 4·27-s − 6·29-s + 2·31-s + 4·33-s − 6·37-s + 2·39-s + 2·41-s + 6·43-s + 45-s − 8·47-s − 7·49-s + 4·51-s − 2·53-s + 2·55-s + 4·57-s + 6·59-s − 14·61-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 1/3·9-s + 0.603·11-s + 0.277·13-s + 0.516·15-s + 0.485·17-s + 0.458·19-s + 0.417·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s + 0.359·31-s + 0.696·33-s − 0.986·37-s + 0.320·39-s + 0.312·41-s + 0.914·43-s + 0.149·45-s − 1.16·47-s − 49-s + 0.560·51-s − 0.274·53-s + 0.269·55-s + 0.529·57-s + 0.781·59-s − 1.79·61-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: yes
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 2.2507080522.250708052
L(12)L(\frac12) ≈\approx 2.2507080522.250708052
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 1−2T+pT2 1 - 2 T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1−2T+pT2 1 - 2 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1+14T+pT2 1 + 14 T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1−10T+pT2 1 - 10 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−2T+pT2 1 - 2 T + p T^{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

−10.79950190094570685551190656629, −9.649255989280370161329944895610, −9.185441066298600697822669255631, −8.282388148593036711889740506020, −7.44095337164983387150060505996, −6.33588991532526860429499225622, −5.24172374409154567149889570954, −3.83609522350488929158971696987, −2.93871105322865322021254198939, −1.63546887865200897104772810976, 1.63546887865200897104772810976, 2.93871105322865322021254198939, 3.83609522350488929158971696987, 5.24172374409154567149889570954, 6.33588991532526860429499225622, 7.44095337164983387150060505996, 8.282388148593036711889740506020, 9.185441066298600697822669255631, 9.649255989280370161329944895610, 10.79950190094570685551190656629

Graph of the ZZ-function along the critical line