↑

Properties

Label 2-588-1.1-c5-0-16
Degree 22
Conductor 588588
Sign −1-1
Analytic cond. 94.305694.3056
Root an. cond. 9.711119.71111
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 9·3-s − 68·5-s + 81·9-s − 388·11-s + 316·13-s + 612·15-s + 1.05e3·17-s − 1.05e3·19-s + 624·23-s + 1.49e3·25-s − 729·27-s + 7.25e3·29-s − 2.29e3·31-s + 3.49e3·33-s + 1.24e4·37-s − 2.84e3·39-s − 5.37e3·41-s + 1.41e4·43-s − 5.50e3·45-s − 4.71e3·47-s − 9.50e3·51-s + 3.78e3·53-s + 2.63e4·55-s + 9.46e3·57-s − 2.52e4·59-s − 2.06e4·61-s − 2.14e4·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.21·5-s + 1/3·9-s − 0.966·11-s + 0.518·13-s + 0.702·15-s + 0.886·17-s − 0.668·19-s + 0.245·23-s + 0.479·25-s − 0.192·27-s + 1.60·29-s − 0.429·31-s + 0.558·33-s + 1.49·37-s − 0.299·39-s − 0.499·41-s + 1.16·43-s − 0.405·45-s − 0.311·47-s − 0.511·51-s + 0.184·53-s + 1.17·55-s + 0.385·57-s − 0.944·59-s − 0.711·61-s − 0.630·65-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: yes
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+p2T 1 + p^{2} T
7 1 1
good5 1+68T+p5T2 1 + 68 T + p^{5} T^{2}
11 1+388T+p5T2 1 + 388 T + p^{5} T^{2}
13 1−316T+p5T2 1 - 316 T + p^{5} T^{2}
17 1−1056T+p5T2 1 - 1056 T + p^{5} T^{2}
19 1+1052T+p5T2 1 + 1052 T + p^{5} T^{2}
23 1−624T+p5T2 1 - 624 T + p^{5} T^{2}
29 1−250pT+p5T2 1 - 250 p T + p^{5} T^{2}
31 1+2296T+p5T2 1 + 2296 T + p^{5} T^{2}
37 1−12426T+p5T2 1 - 12426 T + p^{5} T^{2}
41 1+5376T+p5T2 1 + 5376 T + p^{5} T^{2}
43 1−14164T+p5T2 1 - 14164 T + p^{5} T^{2}
47 1+4712T+p5T2 1 + 4712 T + p^{5} T^{2}
53 1−3782T+p5T2 1 - 3782 T + p^{5} T^{2}
59 1+25244T+p5T2 1 + 25244 T + p^{5} T^{2}
61 1+20668T+p5T2 1 + 20668 T + p^{5} T^{2}
67 1−49012T+p5T2 1 - 49012 T + p^{5} T^{2}
71 1−4760T+p5T2 1 - 4760 T + p^{5} T^{2}
73 1−65264T+p5T2 1 - 65264 T + p^{5} T^{2}
79 1+49736T+p5T2 1 + 49736 T + p^{5} T^{2}
83 1+7788T+p5T2 1 + 7788 T + p^{5} T^{2}
89 1+36904T+p5T2 1 + 36904 T + p^{5} T^{2}
97 1−98264T+p5T2 1 - 98264 T + p^{5} 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

−9.579995150175436473565911838828, −8.280920248418602921874737033106, −7.83770300406472300144127344855, −6.80854968632410975348193474195, −5.78321255851835710278346628820, −4.75721686232100214965746363280, −3.87430414602856007582115479115, −2.73611113748570120100016771361, −1.03333346673946836957456104025, 0, 1.03333346673946836957456104025, 2.73611113748570120100016771361, 3.87430414602856007582115479115, 4.75721686232100214965746363280, 5.78321255851835710278346628820, 6.80854968632410975348193474195, 7.83770300406472300144127344855, 8.280920248418602921874737033106, 9.579995150175436473565911838828

Graph of the ZZ-function along the critical line