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Properties

Label 2-588-1.1-c5-0-29
Degree 22
Conductor 588588
Sign −1-1
Analytic cond. 94.305694.3056
Root an. cond. 9.711119.71111
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·3-s + 96.3·5-s + 81·9-s − 179.·11-s − 56.8·13-s − 867.·15-s + 1.41e3·17-s − 1.29e3·19-s − 4.66e3·23-s + 6.16e3·25-s − 729·27-s − 5.00e3·29-s − 7.44e3·31-s + 1.61e3·33-s − 9.08e3·37-s + 511.·39-s + 6.62e3·41-s − 1.29e4·43-s + 7.80e3·45-s − 5.49e3·47-s − 1.27e4·51-s + 2.95e4·53-s − 1.73e4·55-s + 1.16e4·57-s + 1.05e3·59-s + 9.48e3·61-s − 5.48e3·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.72·5-s + 0.333·9-s − 0.447·11-s − 0.0933·13-s − 0.995·15-s + 1.18·17-s − 0.821·19-s − 1.84·23-s + 1.97·25-s − 0.192·27-s − 1.10·29-s − 1.39·31-s + 0.258·33-s − 1.09·37-s + 0.0538·39-s + 0.615·41-s − 1.06·43-s + 0.574·45-s − 0.363·47-s − 0.686·51-s + 1.44·53-s − 0.771·55-s + 0.474·57-s + 0.0395·59-s + 0.326·61-s − 0.160·65-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, 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: 94.305694.3056
Root analytic conductor: 9.711119.71111
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 588, ( :5/2), −1)(2,\ 588,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{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+9T 1 + 9T
7 1 1
good5 1−96.3T+3.12e3T2 1 - 96.3T + 3.12e3T^{2}
11 1+179.T+1.61e5T2 1 + 179.T + 1.61e5T^{2}
13 1+56.8T+3.71e5T2 1 + 56.8T + 3.71e5T^{2}
17 1−1.41e3T+1.41e6T2 1 - 1.41e3T + 1.41e6T^{2}
19 1+1.29e3T+2.47e6T2 1 + 1.29e3T + 2.47e6T^{2}
23 1+4.66e3T+6.43e6T2 1 + 4.66e3T + 6.43e6T^{2}
29 1+5.00e3T+2.05e7T2 1 + 5.00e3T + 2.05e7T^{2}
31 1+7.44e3T+2.86e7T2 1 + 7.44e3T + 2.86e7T^{2}
37 1+9.08e3T+6.93e7T2 1 + 9.08e3T + 6.93e7T^{2}
41 1−6.62e3T+1.15e8T2 1 - 6.62e3T + 1.15e8T^{2}
43 1+1.29e4T+1.47e8T2 1 + 1.29e4T + 1.47e8T^{2}
47 1+5.49e3T+2.29e8T2 1 + 5.49e3T + 2.29e8T^{2}
53 1−2.95e4T+4.18e8T2 1 - 2.95e4T + 4.18e8T^{2}
59 1−1.05e3T+7.14e8T2 1 - 1.05e3T + 7.14e8T^{2}
61 1−9.48e3T+8.44e8T2 1 - 9.48e3T + 8.44e8T^{2}
67 1−1.33e3T+1.35e9T2 1 - 1.33e3T + 1.35e9T^{2}
71 1−4.98e4T+1.80e9T2 1 - 4.98e4T + 1.80e9T^{2}
73 1+7.57e4T+2.07e9T2 1 + 7.57e4T + 2.07e9T^{2}
79 1−6.23e4T+3.07e9T2 1 - 6.23e4T + 3.07e9T^{2}
83 1−5.32e4T+3.93e9T2 1 - 5.32e4T + 3.93e9T^{2}
89 1+1.18e5T+5.58e9T2 1 + 1.18e5T + 5.58e9T^{2}
97 1+1.36e5T+8.58e9T2 1 + 1.36e5T + 8.58e9T^{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

−9.796135763238285245161693806181, −8.754216189562243225589175078607, −7.59684865624907748689837199291, −6.51696562881039861526819474097, −5.68533177585037145593370268854, −5.26228102428563177642512044190, −3.78251488475464816347125960107, −2.28030353764693032382853695290, −1.53445099053973242533941348184, 0, 1.53445099053973242533941348184, 2.28030353764693032382853695290, 3.78251488475464816347125960107, 5.26228102428563177642512044190, 5.68533177585037145593370268854, 6.51696562881039861526819474097, 7.59684865624907748689837199291, 8.754216189562243225589175078607, 9.796135763238285245161693806181

Graph of the ZZ-function along the critical line