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Properties

Label 2-65-1.1-c5-0-7
Degree 22
Conductor 6565
Sign 11
Analytic cond. 10.424910.4249
Root an. cond. 3.228763.22876
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  − 2.75·2-s + 23.9·3-s − 24.4·4-s − 25·5-s − 66.0·6-s + 85.7·7-s + 155.·8-s + 331.·9-s + 68.9·10-s + 431.·11-s − 584.·12-s + 169·13-s − 236.·14-s − 599.·15-s + 352.·16-s − 438.·17-s − 912.·18-s + 1.61e3·19-s + 610.·20-s + 2.05e3·21-s − 1.18e3·22-s + 2.17e3·23-s + 3.72e3·24-s + 625·25-s − 465.·26-s + 2.11e3·27-s − 2.09e3·28-s + ⋯
L(s)  = 1  − 0.487·2-s + 1.53·3-s − 0.762·4-s − 0.447·5-s − 0.749·6-s + 0.661·7-s + 0.858·8-s + 1.36·9-s + 0.217·10-s + 1.07·11-s − 1.17·12-s + 0.277·13-s − 0.322·14-s − 0.687·15-s + 0.343·16-s − 0.368·17-s − 0.664·18-s + 1.02·19-s + 0.341·20-s + 1.01·21-s − 0.523·22-s + 0.856·23-s + 1.32·24-s + 0.200·25-s − 0.135·26-s + 0.557·27-s − 0.504·28-s + ⋯

Functional equation

Λ(s)=(65s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(65s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6565    =    5⋅135 \cdot 13
Sign: 11
Analytic conductor: 10.424910.4249
Root analytic conductor: 3.228763.22876
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 65, ( :5/2), 1)(2,\ 65,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 2.0134477312.013447731
L(12)L(\frac12) ≈\approx 2.0134477312.013447731
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)
bad5 1+25T 1 + 25T
13 1−169T 1 - 169T
good2 1+2.75T+32T2 1 + 2.75T + 32T^{2}
3 1−23.9T+243T2 1 - 23.9T + 243T^{2}
7 1−85.7T+1.68e4T2 1 - 85.7T + 1.68e4T^{2}
11 1−431.T+1.61e5T2 1 - 431.T + 1.61e5T^{2}
17 1+438.T+1.41e6T2 1 + 438.T + 1.41e6T^{2}
19 1−1.61e3T+2.47e6T2 1 - 1.61e3T + 2.47e6T^{2}
23 1−2.17e3T+6.43e6T2 1 - 2.17e3T + 6.43e6T^{2}
29 1−8.28e3T+2.05e7T2 1 - 8.28e3T + 2.05e7T^{2}
31 1+2.74e3T+2.86e7T2 1 + 2.74e3T + 2.86e7T^{2}
37 1+2.13e3T+6.93e7T2 1 + 2.13e3T + 6.93e7T^{2}
41 1+1.95e4T+1.15e8T2 1 + 1.95e4T + 1.15e8T^{2}
43 1−8.15e3T+1.47e8T2 1 - 8.15e3T + 1.47e8T^{2}
47 1+1.32e4T+2.29e8T2 1 + 1.32e4T + 2.29e8T^{2}
53 1−1.75e3T+4.18e8T2 1 - 1.75e3T + 4.18e8T^{2}
59 1+1.97e3T+7.14e8T2 1 + 1.97e3T + 7.14e8T^{2}
61 1−4.55e4T+8.44e8T2 1 - 4.55e4T + 8.44e8T^{2}
67 1+1.94e4T+1.35e9T2 1 + 1.94e4T + 1.35e9T^{2}
71 1+6.42e4T+1.80e9T2 1 + 6.42e4T + 1.80e9T^{2}
73 1−1.02e3T+2.07e9T2 1 - 1.02e3T + 2.07e9T^{2}
79 1+1.07e5T+3.07e9T2 1 + 1.07e5T + 3.07e9T^{2}
83 1+4.64e4T+3.93e9T2 1 + 4.64e4T + 3.93e9T^{2}
89 1+3.41e3T+5.58e9T2 1 + 3.41e3T + 5.58e9T^{2}
97 1−1.33e5T+8.58e9T2 1 - 1.33e5T + 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

−14.10634898878246715999983386441, −13.14335344102083375388512153942, −11.61364087843365467244319298526, −10.02556916422162250388607798312, −8.895536632513856232301896720900, −8.394952105535746802981364620725, −7.21930438897657010688068794210, −4.64829563190216059202619554769, −3.39033522329111310498604656107, −1.34248783356685451830123653619, 1.34248783356685451830123653619, 3.39033522329111310498604656107, 4.64829563190216059202619554769, 7.21930438897657010688068794210, 8.394952105535746802981364620725, 8.895536632513856232301896720900, 10.02556916422162250388607798312, 11.61364087843365467244319298526, 13.14335344102083375388512153942, 14.10634898878246715999983386441

Graph of the ZZ-function along the critical line