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Properties

Label 2-600-1.1-c5-0-23
Degree 22
Conductor 600600
Sign 11
Analytic cond. 96.230296.2302
Root an. cond. 9.809709.80970
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s + 80·7-s + 81·9-s + 684·11-s + 978·13-s + 862·17-s + 916·19-s + 720·21-s + 1.55e3·23-s + 729·27-s − 7.31e3·29-s − 9.31e3·31-s + 6.15e3·33-s + 8.82e3·37-s + 8.80e3·39-s − 3.28e3·41-s − 7.55e3·43-s + 5.96e3·47-s − 1.04e4·49-s + 7.75e3·51-s + 8.69e3·53-s + 8.24e3·57-s − 4.20e4·59-s + 3.75e4·61-s + 6.48e3·63-s − 2.93e4·67-s + 1.39e4·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.617·7-s + 1/3·9-s + 1.70·11-s + 1.60·13-s + 0.723·17-s + 0.582·19-s + 0.356·21-s + 0.611·23-s + 0.192·27-s − 1.61·29-s − 1.74·31-s + 0.984·33-s + 1.05·37-s + 0.926·39-s − 0.305·41-s − 0.623·43-s + 0.393·47-s − 0.619·49-s + 0.417·51-s + 0.425·53-s + 0.336·57-s − 1.57·59-s + 1.29·61-s + 0.205·63-s − 0.798·67-s + 0.353·69-s + ⋯

Functional equation

Λ(s)=(600s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(600s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 600600    =    23⋅3⋅522^{3} \cdot 3 \cdot 5^{2}
Sign: 11
Analytic conductor: 96.230296.2302
Root analytic conductor: 9.809709.80970
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 600, ( :5/2), 1)(2,\ 600,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 4.0182331454.018233145
L(12)L(\frac12) ≈\approx 4.0182331454.018233145
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
5 1 1
good7 1−80T+p5T2 1 - 80 T + p^{5} T^{2}
11 1−684T+p5T2 1 - 684 T + p^{5} T^{2}
13 1−978T+p5T2 1 - 978 T + p^{5} T^{2}
17 1−862T+p5T2 1 - 862 T + p^{5} T^{2}
19 1−916T+p5T2 1 - 916 T + p^{5} T^{2}
23 1−1552T+p5T2 1 - 1552 T + p^{5} T^{2}
29 1+7314T+p5T2 1 + 7314 T + p^{5} T^{2}
31 1+9312T+p5T2 1 + 9312 T + p^{5} T^{2}
37 1−8826T+p5T2 1 - 8826 T + p^{5} T^{2}
41 1+3286T+p5T2 1 + 3286 T + p^{5} T^{2}
43 1+7556T+p5T2 1 + 7556 T + p^{5} T^{2}
47 1−5960T+p5T2 1 - 5960 T + p^{5} T^{2}
53 1−8698T+p5T2 1 - 8698 T + p^{5} T^{2}
59 1+42036T+p5T2 1 + 42036 T + p^{5} T^{2}
61 1−37518T+p5T2 1 - 37518 T + p^{5} T^{2}
67 1+29324T+p5T2 1 + 29324 T + p^{5} T^{2}
71 1−84408T+p5T2 1 - 84408 T + p^{5} T^{2}
73 1−46550T+p5T2 1 - 46550 T + p^{5} T^{2}
79 1−26752T+p5T2 1 - 26752 T + p^{5} T^{2}
83 1−7956T+p5T2 1 - 7956 T + p^{5} T^{2}
89 1−59674T+p5T2 1 - 59674 T + p^{5} T^{2}
97 1+136898T+p5T2 1 + 136898 T + p^{5} 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

−9.561837485842592351015719079478, −9.083783281079767277602859281337, −8.209718034351718817922422889662, −7.31411826938067185819851643395, −6.32692849030153219449589783202, −5.32694473408731700339080377970, −3.93610865555768492360179681899, −3.45598110410793175982515838798, −1.75984338237498964797833473259, −1.07593525641134169258923820977, 1.07593525641134169258923820977, 1.75984338237498964797833473259, 3.45598110410793175982515838798, 3.93610865555768492360179681899, 5.32694473408731700339080377970, 6.32692849030153219449589783202, 7.31411826938067185819851643395, 8.209718034351718817922422889662, 9.083783281079767277602859281337, 9.561837485842592351015719079478

Graph of the ZZ-function along the critical line