↑

Properties

Label 2-825-1.1-c5-0-121
Degree 22
Conductor 825825
Sign −1-1
Analytic cond. 132.316132.316
Root an. cond. 11.502811.5028
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.33·2-s + 9·3-s − 26.5·4-s + 20.9·6-s − 146.·7-s − 136.·8-s + 81·9-s + 121·11-s − 239.·12-s + 158.·13-s − 341.·14-s + 531.·16-s + 1.92e3·17-s + 188.·18-s − 32.9·19-s − 1.31e3·21-s + 282.·22-s − 2.07e3·23-s − 1.22e3·24-s + 369.·26-s + 729·27-s + 3.88e3·28-s + 1.64e3·29-s − 3.56e3·31-s + 5.61e3·32-s + 1.08e3·33-s + 4.49e3·34-s + ⋯
L(s)  = 1  + 0.412·2-s + 0.577·3-s − 0.829·4-s + 0.238·6-s − 1.12·7-s − 0.754·8-s + 0.333·9-s + 0.301·11-s − 0.479·12-s + 0.259·13-s − 0.465·14-s + 0.518·16-s + 1.61·17-s + 0.137·18-s − 0.0209·19-s − 0.651·21-s + 0.124·22-s − 0.818·23-s − 0.435·24-s + 0.107·26-s + 0.192·27-s + 0.936·28-s + 0.362·29-s − 0.666·31-s + 0.968·32-s + 0.174·33-s + 0.666·34-s + ⋯

Functional equation

Λ(s)=(825s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(825s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, 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: 132.316132.316
Root analytic conductor: 11.502811.5028
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 825, ( :5/2), −1)(2,\ 825,\ (\ :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)
bad3 1−9T 1 - 9T
5 1 1
11 1−121T 1 - 121T
good2 1−2.33T+32T2 1 - 2.33T + 32T^{2}
7 1+146.T+1.68e4T2 1 + 146.T + 1.68e4T^{2}
13 1−158.T+3.71e5T2 1 - 158.T + 3.71e5T^{2}
17 1−1.92e3T+1.41e6T2 1 - 1.92e3T + 1.41e6T^{2}
19 1+32.9T+2.47e6T2 1 + 32.9T + 2.47e6T^{2}
23 1+2.07e3T+6.43e6T2 1 + 2.07e3T + 6.43e6T^{2}
29 1−1.64e3T+2.05e7T2 1 - 1.64e3T + 2.05e7T^{2}
31 1+3.56e3T+2.86e7T2 1 + 3.56e3T + 2.86e7T^{2}
37 1−5.39e3T+6.93e7T2 1 - 5.39e3T + 6.93e7T^{2}
41 1−2.80e3T+1.15e8T2 1 - 2.80e3T + 1.15e8T^{2}
43 1−1.14e3T+1.47e8T2 1 - 1.14e3T + 1.47e8T^{2}
47 1−1.57e4T+2.29e8T2 1 - 1.57e4T + 2.29e8T^{2}
53 1+2.40e4T+4.18e8T2 1 + 2.40e4T + 4.18e8T^{2}
59 1−5.00e3T+7.14e8T2 1 - 5.00e3T + 7.14e8T^{2}
61 1+4.79e3T+8.44e8T2 1 + 4.79e3T + 8.44e8T^{2}
67 1+3.41e4T+1.35e9T2 1 + 3.41e4T + 1.35e9T^{2}
71 1−1.00e4T+1.80e9T2 1 - 1.00e4T + 1.80e9T^{2}
73 1+7.17e4T+2.07e9T2 1 + 7.17e4T + 2.07e9T^{2}
79 1+1.43e4T+3.07e9T2 1 + 1.43e4T + 3.07e9T^{2}
83 1+6.64e4T+3.93e9T2 1 + 6.64e4T + 3.93e9T^{2}
89 1+1.11e4T+5.58e9T2 1 + 1.11e4T + 5.58e9T^{2}
97 1+5.65e4T+8.58e9T2 1 + 5.65e4T + 8.58e9T^{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

−9.188946236445906319574153345098, −8.281068766249702067588085710528, −7.42998234661260713286057856536, −6.23486758531879141882411999205, −5.57593871869554587041963759520, −4.31484913506695765515812008384, −3.55307316552815280205947802213, −2.84914741973679579029776514844, −1.21691700057161989784813509351, 0, 1.21691700057161989784813509351, 2.84914741973679579029776514844, 3.55307316552815280205947802213, 4.31484913506695765515812008384, 5.57593871869554587041963759520, 6.23486758531879141882411999205, 7.42998234661260713286057856536, 8.281068766249702067588085710528, 9.188946236445906319574153345098

Graph of the ZZ-function along the critical line