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Properties

Label 2-210-1.1-c5-0-17
Degree 22
Conductor 210210
Sign −1-1
Analytic cond. 33.680633.6806
Root an. cond. 5.803495.80349
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 9·3-s + 16·4-s − 25·5-s − 36·6-s + 49·7-s + 64·8-s + 81·9-s − 100·10-s − 492·11-s − 144·12-s + 938·13-s + 196·14-s + 225·15-s + 256·16-s − 1.93e3·17-s + 324·18-s + 248·19-s − 400·20-s − 441·21-s − 1.96e3·22-s + 60·23-s − 576·24-s + 625·25-s + 3.75e3·26-s − 729·27-s + 784·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.22·11-s − 0.288·12-s + 1.53·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.62·17-s + 0.235·18-s + 0.157·19-s − 0.223·20-s − 0.218·21-s − 0.866·22-s + 0.0236·23-s − 0.204·24-s + 1/5·25-s + 1.08·26-s − 0.192·27-s + 0.188·28-s + ⋯

Functional equation

Λ(s)=(210s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(210s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 210210    =    2⋅3⋅5⋅72 \cdot 3 \cdot 5 \cdot 7
Sign: −1-1
Analytic conductor: 33.680633.6806
Root analytic conductor: 5.803495.80349
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 210, ( :5/2), −1)(2,\ 210,\ (\ :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−p2T 1 - p^{2} T
3 1+p2T 1 + p^{2} T
5 1+p2T 1 + p^{2} T
7 1−p2T 1 - p^{2} T
good11 1+492T+p5T2 1 + 492 T + p^{5} T^{2}
13 1−938T+p5T2 1 - 938 T + p^{5} T^{2}
17 1+114pT+p5T2 1 + 114 p T + p^{5} T^{2}
19 1−248T+p5T2 1 - 248 T + p^{5} T^{2}
23 1−60T+p5T2 1 - 60 T + p^{5} T^{2}
29 1+6546T+p5T2 1 + 6546 T + p^{5} T^{2}
31 1+3484T+p5T2 1 + 3484 T + p^{5} T^{2}
37 1+2938T+p5T2 1 + 2938 T + p^{5} T^{2}
41 1+17754T+p5T2 1 + 17754 T + p^{5} T^{2}
43 1+9748T+p5T2 1 + 9748 T + p^{5} T^{2}
47 1−7656T+p5T2 1 - 7656 T + p^{5} T^{2}
53 1+21198T+p5T2 1 + 21198 T + p^{5} T^{2}
59 1+10188T+p5T2 1 + 10188 T + p^{5} T^{2}
61 1+1930T+p5T2 1 + 1930 T + p^{5} T^{2}
67 1+22348T+p5T2 1 + 22348 T + p^{5} T^{2}
71 1+2184T+p5T2 1 + 2184 T + p^{5} T^{2}
73 1+53410T+p5T2 1 + 53410 T + p^{5} T^{2}
79 1+44896T+p5T2 1 + 44896 T + p^{5} T^{2}
83 1−84324T+p5T2 1 - 84324 T + p^{5} T^{2}
89 1−130950T+p5T2 1 - 130950 T + p^{5} T^{2}
97 1−156830T+p5T2 1 - 156830 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

−11.06334050104845286509689993111, −10.58586358385969120935629006063, −8.889558343079709824040191035818, −7.81247868816288985539895225323, −6.70152200483342690857868169703, −5.61705972764294132422848289779, −4.63437734101092637367893792983, −3.47035688322890505204673619302, −1.81416228489110967430133918984, 0, 1.81416228489110967430133918984, 3.47035688322890505204673619302, 4.63437734101092637367893792983, 5.61705972764294132422848289779, 6.70152200483342690857868169703, 7.81247868816288985539895225323, 8.889558343079709824040191035818, 10.58586358385969120935629006063, 11.06334050104845286509689993111

Graph of the ZZ-function along the critical line