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Properties

Label 2-825-1.1-c3-0-70
Degree 22
Conductor 825825
Sign −1-1
Analytic cond. 48.676548.6765
Root an. cond. 6.976866.97686
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.723·2-s + 3·3-s − 7.47·4-s − 2.17·6-s + 1.13·7-s + 11.1·8-s + 9·9-s + 11·11-s − 22.4·12-s − 21.8·13-s − 0.819·14-s + 51.7·16-s + 6.18·17-s − 6.51·18-s − 92.8·19-s + 3.39·21-s − 7.96·22-s − 36.7·23-s + 33.5·24-s + 15.8·26-s + 27·27-s − 8.46·28-s + 71.1·29-s + 186.·31-s − 127.·32-s + 33·33-s − 4.47·34-s + ⋯
L(s)  = 1  − 0.255·2-s + 0.577·3-s − 0.934·4-s − 0.147·6-s + 0.0611·7-s + 0.494·8-s + 0.333·9-s + 0.301·11-s − 0.539·12-s − 0.466·13-s − 0.0156·14-s + 0.807·16-s + 0.0881·17-s − 0.0852·18-s − 1.12·19-s + 0.0353·21-s − 0.0771·22-s − 0.333·23-s + 0.285·24-s + 0.119·26-s + 0.192·27-s − 0.0571·28-s + 0.455·29-s + 1.08·31-s − 0.701·32-s + 0.174·33-s − 0.0225·34-s + ⋯

Functional equation

Λ(s)=(825s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(825s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 825825    =    3⋅52⋅113 \cdot 5^{2} \cdot 11
Sign: −1-1
Analytic conductor: 48.676548.6765
Root analytic conductor: 6.976866.97686
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 825, ( :3/2), −1)(2,\ 825,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad3 1−3T 1 - 3T
5 1 1
11 1−11T 1 - 11T
good2 1+0.723T+8T2 1 + 0.723T + 8T^{2}
7 1−1.13T+343T2 1 - 1.13T + 343T^{2}
13 1+21.8T+2.19e3T2 1 + 21.8T + 2.19e3T^{2}
17 1−6.18T+4.91e3T2 1 - 6.18T + 4.91e3T^{2}
19 1+92.8T+6.85e3T2 1 + 92.8T + 6.85e3T^{2}
23 1+36.7T+1.21e4T2 1 + 36.7T + 1.21e4T^{2}
29 1−71.1T+2.43e4T2 1 - 71.1T + 2.43e4T^{2}
31 1−186.T+2.97e4T2 1 - 186.T + 2.97e4T^{2}
37 1+356.T+5.06e4T2 1 + 356.T + 5.06e4T^{2}
41 1−271.T+6.89e4T2 1 - 271.T + 6.89e4T^{2}
43 1−155.T+7.95e4T2 1 - 155.T + 7.95e4T^{2}
47 1−234.T+1.03e5T2 1 - 234.T + 1.03e5T^{2}
53 1−195.T+1.48e5T2 1 - 195.T + 1.48e5T^{2}
59 1+455.T+2.05e5T2 1 + 455.T + 2.05e5T^{2}
61 1+441.T+2.26e5T2 1 + 441.T + 2.26e5T^{2}
67 1−133.T+3.00e5T2 1 - 133.T + 3.00e5T^{2}
71 1+1.04e3T+3.57e5T2 1 + 1.04e3T + 3.57e5T^{2}
73 1+160.T+3.89e5T2 1 + 160.T + 3.89e5T^{2}
79 1+761.T+4.93e5T2 1 + 761.T + 4.93e5T^{2}
83 1−51.7T+5.71e5T2 1 - 51.7T + 5.71e5T^{2}
89 1+1.07e3T+7.04e5T2 1 + 1.07e3T + 7.04e5T^{2}
97 1+703.T+9.12e5T2 1 + 703.T + 9.12e5T^{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

−9.303096338694315331886966347972, −8.616664110169027445670224872884, −7.962484579762390112228576164610, −6.99754085167477805901229589265, −5.85866678728864268153075339313, −4.66974320855044842980915030680, −4.03340274172898315513678322146, −2.78583340557904427382802754752, −1.43239054369412809949472506116, 0, 1.43239054369412809949472506116, 2.78583340557904427382802754752, 4.03340274172898315513678322146, 4.66974320855044842980915030680, 5.85866678728864268153075339313, 6.99754085167477805901229589265, 7.962484579762390112228576164610, 8.616664110169027445670224872884, 9.303096338694315331886966347972

Graph of the ZZ-function along the critical line