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Properties

Label 2-588-1.1-c1-0-5
Degree 22
Conductor 588588
Sign −1-1
Analytic cond. 4.695204.69520
Root an. cond. 2.166842.16684
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  − 3-s + 9-s − 6·11-s − 2·13-s + 4·19-s − 6·23-s − 5·25-s − 27-s + 6·29-s − 8·31-s + 6·33-s + 2·37-s + 2·39-s − 12·41-s − 4·43-s − 12·47-s − 6·53-s − 4·57-s + 10·61-s + 8·67-s + 6·69-s + 6·71-s + 10·73-s + 5·75-s − 4·79-s + 81-s + 12·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.80·11-s − 0.554·13-s + 0.917·19-s − 1.25·23-s − 25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 1.04·33-s + 0.328·37-s + 0.320·39-s − 1.87·41-s − 0.609·43-s − 1.75·47-s − 0.824·53-s − 0.529·57-s + 1.28·61-s + 0.977·67-s + 0.722·69-s + 0.712·71-s + 1.17·73-s + 0.577·75-s − 0.450·79-s + 1/9·81-s + 1.31·83-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: −1-1
Analytic conductor: 4.695204.69520
Root analytic conductor: 2.166842.16684
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 588, ( :1/2), −1)(2,\ 588,\ (\ :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
3 1+T 1 + T
7 1 1
good5 1+pT2 1 + p T^{2}
11 1+6T+pT2 1 + 6 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+12T+pT2 1 + 12 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1−10T+pT2 1 - 10 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+12T+pT2 1 + 12 T + p T^{2}
97 1−10T+pT2 1 - 10 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

−10.11620249108594953836248897914, −9.790967618641863520242547413629, −8.245440388686575346900612009742, −7.67615829141164438776279339720, −6.60712751651675961597255577118, −5.45823535038952231993330195200, −4.92172543859600215194027340087, −3.44705194784191488479832061991, −2.08973766081077835946990875259, 0, 2.08973766081077835946990875259, 3.44705194784191488479832061991, 4.92172543859600215194027340087, 5.45823535038952231993330195200, 6.60712751651675961597255577118, 7.67615829141164438776279339720, 8.245440388686575346900612009742, 9.790967618641863520242547413629, 10.11620249108594953836248897914

Graph of the ZZ-function along the critical line