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Properties

Label 2-520-1.1-c1-0-7
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.44·3-s + 5-s + 2·7-s + 2.99·9-s + 0.449·11-s − 13-s + 2.44·15-s − 2.89·17-s − 4.44·19-s + 4.89·21-s + 1.55·23-s + 25-s + 4·29-s + 0.449·31-s + 1.10·33-s + 2·35-s − 4.89·37-s − 2.44·39-s + 1.10·41-s − 3.34·43-s + 2.99·45-s − 2·47-s − 3·49-s − 7.10·51-s + 10.8·53-s + 0.449·55-s − 10.8·57-s + ⋯
L(s)  = 1  + 1.41·3-s + 0.447·5-s + 0.755·7-s + 0.999·9-s + 0.135·11-s − 0.277·13-s + 0.632·15-s − 0.703·17-s − 1.02·19-s + 1.06·21-s + 0.323·23-s + 0.200·25-s + 0.742·29-s + 0.0807·31-s + 0.191·33-s + 0.338·35-s − 0.805·37-s − 0.392·39-s + 0.171·41-s − 0.510·43-s + 0.447·45-s − 0.291·47-s − 0.428·49-s − 0.994·51-s + 1.49·53-s + 0.0606·55-s − 1.44·57-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 2.4736735982.473673598
L(12)L(\frac12) ≈\approx 2.4736735982.473673598
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 1−2.44T+3T2 1 - 2.44T + 3T^{2}
7 1−2T+7T2 1 - 2T + 7T^{2}
11 1−0.449T+11T2 1 - 0.449T + 11T^{2}
17 1+2.89T+17T2 1 + 2.89T + 17T^{2}
19 1+4.44T+19T2 1 + 4.44T + 19T^{2}
23 1−1.55T+23T2 1 - 1.55T + 23T^{2}
29 1−4T+29T2 1 - 4T + 29T^{2}
31 1−0.449T+31T2 1 - 0.449T + 31T^{2}
37 1+4.89T+37T2 1 + 4.89T + 37T^{2}
41 1−1.10T+41T2 1 - 1.10T + 41T^{2}
43 1+3.34T+43T2 1 + 3.34T + 43T^{2}
47 1+2T+47T2 1 + 2T + 47T^{2}
53 1−10.8T+53T2 1 - 10.8T + 53T^{2}
59 1+5.34T+59T2 1 + 5.34T + 59T^{2}
61 1−13.7T+61T2 1 - 13.7T + 61T^{2}
67 1+14.8T+67T2 1 + 14.8T + 67T^{2}
71 1+8.44T+71T2 1 + 8.44T + 71T^{2}
73 1−14.6T+73T2 1 - 14.6T + 73T^{2}
79 1+4.89T+79T2 1 + 4.89T + 79T^{2}
83 1−2T+83T2 1 - 2T + 83T^{2}
89 1−6T+89T2 1 - 6T + 89T^{2}
97 1+11.7T+97T2 1 + 11.7T + 97T^{2}
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   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

−10.70525758672027089757299013829, −9.855490631365725808039136213883, −8.828004100374299851941654892946, −8.468772761594242766495781029739, −7.45021289498554912174980419510, −6.45042225654760342330317057616, −5.04198320919738406846616321643, −4.01831226697534092421927954877, −2.72510253910502481469015184697, −1.80293423917785422604569313169, 1.80293423917785422604569313169, 2.72510253910502481469015184697, 4.01831226697534092421927954877, 5.04198320919738406846616321643, 6.45042225654760342330317057616, 7.45021289498554912174980419510, 8.468772761594242766495781029739, 8.828004100374299851941654892946, 9.855490631365725808039136213883, 10.70525758672027089757299013829

Graph of the ZZ-function along the critical line