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Properties

Label 2-650-1.1-c5-0-67
Degree 22
Conductor 650650
Sign −1-1
Analytic cond. 104.249104.249
Root an. cond. 10.210210.2102
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  + 4·2-s − 15.0·3-s + 16·4-s − 60.1·6-s − 52.9·7-s + 64·8-s − 16.7·9-s − 259.·11-s − 240.·12-s + 169·13-s − 211.·14-s + 256·16-s + 2.27e3·17-s − 67.0·18-s + 730.·19-s + 796.·21-s − 1.03e3·22-s − 1.97e3·23-s − 962.·24-s + 676·26-s + 3.90e3·27-s − 847.·28-s − 949.·29-s + 225.·31-s + 1.02e3·32-s + 3.89e3·33-s + 9.09e3·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.964·3-s + 0.5·4-s − 0.682·6-s − 0.408·7-s + 0.353·8-s − 0.0689·9-s − 0.646·11-s − 0.482·12-s + 0.277·13-s − 0.288·14-s + 0.250·16-s + 1.90·17-s − 0.0487·18-s + 0.464·19-s + 0.394·21-s − 0.456·22-s − 0.777·23-s − 0.341·24-s + 0.196·26-s + 1.03·27-s − 0.204·28-s − 0.209·29-s + 0.0421·31-s + 0.176·32-s + 0.623·33-s + 1.34·34-s + ⋯

Functional equation

Λ(s)=(650s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(650s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 650650    =    2⋅52⋅132 \cdot 5^{2} \cdot 13
Sign: −1-1
Analytic conductor: 104.249104.249
Root analytic conductor: 10.210210.2102
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 650, ( :5/2), −1)(2,\ 650,\ (\ :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)
bad2 1−4T 1 - 4T
5 1 1
13 1−169T 1 - 169T
good3 1+15.0T+243T2 1 + 15.0T + 243T^{2}
7 1+52.9T+1.68e4T2 1 + 52.9T + 1.68e4T^{2}
11 1+259.T+1.61e5T2 1 + 259.T + 1.61e5T^{2}
17 1−2.27e3T+1.41e6T2 1 - 2.27e3T + 1.41e6T^{2}
19 1−730.T+2.47e6T2 1 - 730.T + 2.47e6T^{2}
23 1+1.97e3T+6.43e6T2 1 + 1.97e3T + 6.43e6T^{2}
29 1+949.T+2.05e7T2 1 + 949.T + 2.05e7T^{2}
31 1−225.T+2.86e7T2 1 - 225.T + 2.86e7T^{2}
37 1−954.T+6.93e7T2 1 - 954.T + 6.93e7T^{2}
41 1+1.73e4T+1.15e8T2 1 + 1.73e4T + 1.15e8T^{2}
43 1−1.00e4T+1.47e8T2 1 - 1.00e4T + 1.47e8T^{2}
47 1−6.06e3T+2.29e8T2 1 - 6.06e3T + 2.29e8T^{2}
53 1−1.61e4T+4.18e8T2 1 - 1.61e4T + 4.18e8T^{2}
59 1−326.T+7.14e8T2 1 - 326.T + 7.14e8T^{2}
61 1+4.68e4T+8.44e8T2 1 + 4.68e4T + 8.44e8T^{2}
67 1+4.35e4T+1.35e9T2 1 + 4.35e4T + 1.35e9T^{2}
71 1−5.17e4T+1.80e9T2 1 - 5.17e4T + 1.80e9T^{2}
73 1−1.32e4T+2.07e9T2 1 - 1.32e4T + 2.07e9T^{2}
79 1−7.62e4T+3.07e9T2 1 - 7.62e4T + 3.07e9T^{2}
83 1+3.01e4T+3.93e9T2 1 + 3.01e4T + 3.93e9T^{2}
89 1+6.67e4T+5.58e9T2 1 + 6.67e4T + 5.58e9T^{2}
97 1+1.51e5T+8.58e9T2 1 + 1.51e5T + 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

−9.661424740767106877540296716837, −8.257915936484532617406275530042, −7.41391273703740005334771461912, −6.32309051944198239145421325964, −5.63810298597729792091611634647, −5.03642640504899894289498140108, −3.70950936553542139256605393018, −2.80484413018033340477116498766, −1.25346453166362685586256091356, 0, 1.25346453166362685586256091356, 2.80484413018033340477116498766, 3.70950936553542139256605393018, 5.03642640504899894289498140108, 5.63810298597729792091611634647, 6.32309051944198239145421325964, 7.41391273703740005334771461912, 8.257915936484532617406275530042, 9.661424740767106877540296716837

Graph of the ZZ-function along the critical line