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Properties

Label 2-825-1.1-c5-0-58
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  + 8.07·2-s − 9·3-s + 33.2·4-s − 72.7·6-s + 39.3·7-s + 10.2·8-s + 81·9-s + 121·11-s − 299.·12-s + 220.·13-s + 318.·14-s − 981.·16-s + 200.·17-s + 654.·18-s + 350.·19-s − 354.·21-s + 977.·22-s − 1.38e3·23-s − 91.9·24-s + 1.78e3·26-s − 729·27-s + 1.30e3·28-s + 5.50e3·29-s − 2.45e3·31-s − 8.25e3·32-s − 1.08e3·33-s + 1.61e3·34-s + ⋯
L(s)  = 1  + 1.42·2-s − 0.577·3-s + 1.03·4-s − 0.824·6-s + 0.303·7-s + 0.0564·8-s + 0.333·9-s + 0.301·11-s − 0.600·12-s + 0.362·13-s + 0.433·14-s − 0.958·16-s + 0.168·17-s + 0.476·18-s + 0.222·19-s − 0.175·21-s + 0.430·22-s − 0.545·23-s − 0.0325·24-s + 0.517·26-s − 0.192·27-s + 0.315·28-s + 1.21·29-s − 0.458·31-s − 1.42·32-s − 0.174·33-s + 0.240·34-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 4.1864929614.186492961
L(12)L(\frac12) ≈\approx 4.1864929614.186492961
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 1−8.07T+32T2 1 - 8.07T + 32T^{2}
7 1−39.3T+1.68e4T2 1 - 39.3T + 1.68e4T^{2}
13 1−220.T+3.71e5T2 1 - 220.T + 3.71e5T^{2}
17 1−200.T+1.41e6T2 1 - 200.T + 1.41e6T^{2}
19 1−350.T+2.47e6T2 1 - 350.T + 2.47e6T^{2}
23 1+1.38e3T+6.43e6T2 1 + 1.38e3T + 6.43e6T^{2}
29 1−5.50e3T+2.05e7T2 1 - 5.50e3T + 2.05e7T^{2}
31 1+2.45e3T+2.86e7T2 1 + 2.45e3T + 2.86e7T^{2}
37 1−4.06e3T+6.93e7T2 1 - 4.06e3T + 6.93e7T^{2}
41 1−527.T+1.15e8T2 1 - 527.T + 1.15e8T^{2}
43 1−1.20e4T+1.47e8T2 1 - 1.20e4T + 1.47e8T^{2}
47 1+563.T+2.29e8T2 1 + 563.T + 2.29e8T^{2}
53 1−3.72e4T+4.18e8T2 1 - 3.72e4T + 4.18e8T^{2}
59 1−2.15e3T+7.14e8T2 1 - 2.15e3T + 7.14e8T^{2}
61 1+3.99e4T+8.44e8T2 1 + 3.99e4T + 8.44e8T^{2}
67 1−3.84e4T+1.35e9T2 1 - 3.84e4T + 1.35e9T^{2}
71 1+1.37e4T+1.80e9T2 1 + 1.37e4T + 1.80e9T^{2}
73 1−3.97e4T+2.07e9T2 1 - 3.97e4T + 2.07e9T^{2}
79 1−3.56e4T+3.07e9T2 1 - 3.56e4T + 3.07e9T^{2}
83 1−7.99e4T+3.93e9T2 1 - 7.99e4T + 3.93e9T^{2}
89 1+3.77e4T+5.58e9T2 1 + 3.77e4T + 5.58e9T^{2}
97 1−7.61e3T+8.58e9T2 1 - 7.61e3T + 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

−9.593396194088515146350330895503, −8.580990740455532707459821914283, −7.45922811635295521071980971882, −6.46829463597262933770636335296, −5.85234625320562815312527535114, −4.97774192926619066629652996187, −4.23026222599976917878515372944, −3.33680405051253071304348989548, −2.13069831831111957379794882914, −0.77309700318192769304041189439, 0.77309700318192769304041189439, 2.13069831831111957379794882914, 3.33680405051253071304348989548, 4.23026222599976917878515372944, 4.97774192926619066629652996187, 5.85234625320562815312527535114, 6.46829463597262933770636335296, 7.45922811635295521071980971882, 8.580990740455532707459821914283, 9.593396194088515146350330895503

Graph of the ZZ-function along the critical line