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Properties

Label 2-825-1.1-c5-0-113
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.20·2-s − 9·3-s + 52.7·4-s + 82.8·6-s + 97.7·7-s − 191.·8-s + 81·9-s + 121·11-s − 474.·12-s + 490.·13-s − 900.·14-s + 72.4·16-s + 881.·17-s − 745.·18-s + 34.4·19-s − 880.·21-s − 1.11e3·22-s + 2.90e3·23-s + 1.72e3·24-s − 4.51e3·26-s − 729·27-s + 5.16e3·28-s − 1.41e3·29-s − 2.53e3·31-s + 5.45e3·32-s − 1.08e3·33-s − 8.11e3·34-s + ⋯
L(s)  = 1  − 1.62·2-s − 0.577·3-s + 1.64·4-s + 0.939·6-s + 0.754·7-s − 1.05·8-s + 0.333·9-s + 0.301·11-s − 0.952·12-s + 0.804·13-s − 1.22·14-s + 0.0707·16-s + 0.739·17-s − 0.542·18-s + 0.0218·19-s − 0.435·21-s − 0.490·22-s + 1.14·23-s + 0.610·24-s − 1.30·26-s − 0.192·27-s + 1.24·28-s − 0.311·29-s − 0.473·31-s + 0.941·32-s − 0.174·33-s − 1.20·34-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+9.20T+32T2 1 + 9.20T + 32T^{2}
7 1−97.7T+1.68e4T2 1 - 97.7T + 1.68e4T^{2}
13 1−490.T+3.71e5T2 1 - 490.T + 3.71e5T^{2}
17 1−881.T+1.41e6T2 1 - 881.T + 1.41e6T^{2}
19 1−34.4T+2.47e6T2 1 - 34.4T + 2.47e6T^{2}
23 1−2.90e3T+6.43e6T2 1 - 2.90e3T + 6.43e6T^{2}
29 1+1.41e3T+2.05e7T2 1 + 1.41e3T + 2.05e7T^{2}
31 1+2.53e3T+2.86e7T2 1 + 2.53e3T + 2.86e7T^{2}
37 1+6.26e3T+6.93e7T2 1 + 6.26e3T + 6.93e7T^{2}
41 1+1.82e4T+1.15e8T2 1 + 1.82e4T + 1.15e8T^{2}
43 1+1.41e4T+1.47e8T2 1 + 1.41e4T + 1.47e8T^{2}
47 1−7.81e3T+2.29e8T2 1 - 7.81e3T + 2.29e8T^{2}
53 1−4.61e3T+4.18e8T2 1 - 4.61e3T + 4.18e8T^{2}
59 1+1.12e3T+7.14e8T2 1 + 1.12e3T + 7.14e8T^{2}
61 1+672.T+8.44e8T2 1 + 672.T + 8.44e8T^{2}
67 1+2.92e4T+1.35e9T2 1 + 2.92e4T + 1.35e9T^{2}
71 1+9.69e3T+1.80e9T2 1 + 9.69e3T + 1.80e9T^{2}
73 1+4.03e4T+2.07e9T2 1 + 4.03e4T + 2.07e9T^{2}
79 1+7.28e4T+3.07e9T2 1 + 7.28e4T + 3.07e9T^{2}
83 1+1.00e5T+3.93e9T2 1 + 1.00e5T + 3.93e9T^{2}
89 1+3.06e4T+5.58e9T2 1 + 3.06e4T + 5.58e9T^{2}
97 1−3.44e4T+8.58e9T2 1 - 3.44e4T + 8.58e9T^{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

−8.904299417702351705426714776014, −8.434837339507127217452725639631, −7.44639189706763761985447757552, −6.81383279401287945133259772838, −5.73802724679002358051712229407, −4.71818951313862725266404429035, −3.30979683040344202718298161985, −1.72860520334587632271734111317, −1.16413082909222054865008910520, 0, 1.16413082909222054865008910520, 1.72860520334587632271734111317, 3.30979683040344202718298161985, 4.71818951313862725266404429035, 5.73802724679002358051712229407, 6.81383279401287945133259772838, 7.44639189706763761985447757552, 8.434837339507127217452725639631, 8.904299417702351705426714776014

Graph of the ZZ-function along the critical line