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Properties

Label 2-825-1.1-c5-0-12
Degree 22
Conductor 825825
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.55·2-s + 9·3-s + 11.0·4-s − 59.0·6-s − 146.·7-s + 137.·8-s + 81·9-s − 121·11-s + 99.1·12-s − 170.·13-s + 960.·14-s − 1.25e3·16-s − 1.56e3·17-s − 531.·18-s + 569.·19-s − 1.31e3·21-s + 793.·22-s − 3.15e3·23-s + 1.23e3·24-s + 1.11e3·26-s + 729·27-s − 1.61e3·28-s + 3.98e3·29-s + 2.99e3·31-s + 3.82e3·32-s − 1.08e3·33-s + 1.02e4·34-s + ⋯
L(s)  = 1  − 1.15·2-s + 0.577·3-s + 0.344·4-s − 0.669·6-s − 1.12·7-s + 0.760·8-s + 0.333·9-s − 0.301·11-s + 0.198·12-s − 0.279·13-s + 1.31·14-s − 1.22·16-s − 1.31·17-s − 0.386·18-s + 0.362·19-s − 0.652·21-s + 0.349·22-s − 1.24·23-s + 0.438·24-s + 0.323·26-s + 0.192·27-s − 0.389·28-s + 0.879·29-s + 0.558·31-s + 0.661·32-s − 0.174·33-s + 1.52·34-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: 11
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: 00
Selberg data: (2, 825, ( :5/2), 1)(2,\ 825,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.50106880610.5010688061
L(12)L(\frac12) ≈\approx 0.50106880610.5010688061
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+6.55T+32T2 1 + 6.55T + 32T^{2}
7 1+146.T+1.68e4T2 1 + 146.T + 1.68e4T^{2}
13 1+170.T+3.71e5T2 1 + 170.T + 3.71e5T^{2}
17 1+1.56e3T+1.41e6T2 1 + 1.56e3T + 1.41e6T^{2}
19 1−569.T+2.47e6T2 1 - 569.T + 2.47e6T^{2}
23 1+3.15e3T+6.43e6T2 1 + 3.15e3T + 6.43e6T^{2}
29 1−3.98e3T+2.05e7T2 1 - 3.98e3T + 2.05e7T^{2}
31 1−2.99e3T+2.86e7T2 1 - 2.99e3T + 2.86e7T^{2}
37 1+7.85e3T+6.93e7T2 1 + 7.85e3T + 6.93e7T^{2}
41 1+5.20e3T+1.15e8T2 1 + 5.20e3T + 1.15e8T^{2}
43 1+1.38e4T+1.47e8T2 1 + 1.38e4T + 1.47e8T^{2}
47 1+6.85e3T+2.29e8T2 1 + 6.85e3T + 2.29e8T^{2}
53 1−3.83e3T+4.18e8T2 1 - 3.83e3T + 4.18e8T^{2}
59 1−9.64e3T+7.14e8T2 1 - 9.64e3T + 7.14e8T^{2}
61 1+2.11e4T+8.44e8T2 1 + 2.11e4T + 8.44e8T^{2}
67 1−4.34e4T+1.35e9T2 1 - 4.34e4T + 1.35e9T^{2}
71 1+5.26e4T+1.80e9T2 1 + 5.26e4T + 1.80e9T^{2}
73 1−6.43e4T+2.07e9T2 1 - 6.43e4T + 2.07e9T^{2}
79 1−2.89e4T+3.07e9T2 1 - 2.89e4T + 3.07e9T^{2}
83 1−4.64e3T+3.93e9T2 1 - 4.64e3T + 3.93e9T^{2}
89 1+1.03e5T+5.58e9T2 1 + 1.03e5T + 5.58e9T^{2}
97 1+7.58e4T+8.58e9T2 1 + 7.58e4T + 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

−9.542745238201600010050308100414, −8.626471160625154004603973785007, −8.102718313577529306531088851740, −7.04229408658193004512435942869, −6.43393259516918565755940309837, −4.95152447542382480304315965095, −3.90476392203606472018964619067, −2.75814671586818336309329274646, −1.75103608849350461757826734523, −0.36346721324190111573002025139, 0.36346721324190111573002025139, 1.75103608849350461757826734523, 2.75814671586818336309329274646, 3.90476392203606472018964619067, 4.95152447542382480304315965095, 6.43393259516918565755940309837, 7.04229408658193004512435942869, 8.102718313577529306531088851740, 8.626471160625154004603973785007, 9.542745238201600010050308100414

Graph of the ZZ-function along the critical line