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Properties

Label 2-520-1.1-c1-0-10
Degree 22
Conductor 520520
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 3·9-s − 4·11-s − 13-s − 6·17-s + 4·19-s + 25-s − 2·29-s − 4·31-s − 6·37-s − 6·41-s + 8·43-s + 3·45-s − 7·49-s + 2·53-s + 4·55-s + 4·59-s − 10·61-s + 65-s + 12·67-s − 4·71-s + 14·73-s − 16·79-s + 9·81-s + 12·83-s + 6·85-s + 2·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 9-s − 1.20·11-s − 0.277·13-s − 1.45·17-s + 0.917·19-s + 1/5·25-s − 0.371·29-s − 0.718·31-s − 0.986·37-s − 0.937·41-s + 1.21·43-s + 0.447·45-s − 49-s + 0.274·53-s + 0.539·55-s + 0.520·59-s − 1.28·61-s + 0.124·65-s + 1.46·67-s − 0.474·71-s + 1.63·73-s − 1.80·79-s + 81-s + 1.31·83-s + 0.650·85-s + 0.211·89-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: −1-1
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: 11
Selberg data: (2, 520, ( :1/2), −1)(2,\ 520,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+pT2 1 + p T^{2}
7 1+pT2 1 + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1+4T+pT2 1 + 4 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1+16T+pT2 1 + 16 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−2T+pT2 1 - 2 T + p T^{2}
97 1+2T+pT2 1 + 2 T + p T^{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

−10.64922295335439854088297490672, −9.462643365908751567949324738666, −8.594892869331333585303013523541, −7.79079474566761964794309748173, −6.86780410309443942051298695606, −5.62737400849428477363472437992, −4.82936854271082841274403128466, −3.43584232426055948992159380385, −2.33964853650557930856432843128, 0, 2.33964853650557930856432843128, 3.43584232426055948992159380385, 4.82936854271082841274403128466, 5.62737400849428477363472437992, 6.86780410309443942051298695606, 7.79079474566761964794309748173, 8.594892869331333585303013523541, 9.462643365908751567949324738666, 10.64922295335439854088297490672

Graph of the ZZ-function along the critical line