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Properties

Label 2-650-1.1-c5-0-76
Degree 22
Conductor 650650
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 27.8·3-s + 16·4-s + 111.·6-s + 240.·7-s + 64·8-s + 534.·9-s + 544.·11-s + 446.·12-s − 169·13-s + 961.·14-s + 256·16-s − 1.62e3·17-s + 2.13e3·18-s − 805.·19-s + 6.70e3·21-s + 2.17e3·22-s − 373.·23-s + 1.78e3·24-s − 676·26-s + 8.14e3·27-s + 3.84e3·28-s + 1.50e3·29-s − 2.20e3·31-s + 1.02e3·32-s + 1.51e4·33-s − 6.51e3·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.78·3-s + 0.5·4-s + 1.26·6-s + 1.85·7-s + 0.353·8-s + 2.20·9-s + 1.35·11-s + 0.894·12-s − 0.277·13-s + 1.31·14-s + 0.250·16-s − 1.36·17-s + 1.55·18-s − 0.512·19-s + 3.31·21-s + 0.958·22-s − 0.147·23-s + 0.632·24-s − 0.196·26-s + 2.14·27-s + 0.927·28-s + 0.332·29-s − 0.411·31-s + 0.176·32-s + 2.42·33-s − 0.966·34-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: 11
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: 00
Selberg data: (2, 650, ( :5/2), 1)(2,\ 650,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 9.4481244979.448124497
L(12)L(\frac12) ≈\approx 9.4481244979.448124497
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 1−27.8T+243T2 1 - 27.8T + 243T^{2}
7 1−240.T+1.68e4T2 1 - 240.T + 1.68e4T^{2}
11 1−544.T+1.61e5T2 1 - 544.T + 1.61e5T^{2}
17 1+1.62e3T+1.41e6T2 1 + 1.62e3T + 1.41e6T^{2}
19 1+805.T+2.47e6T2 1 + 805.T + 2.47e6T^{2}
23 1+373.T+6.43e6T2 1 + 373.T + 6.43e6T^{2}
29 1−1.50e3T+2.05e7T2 1 - 1.50e3T + 2.05e7T^{2}
31 1+2.20e3T+2.86e7T2 1 + 2.20e3T + 2.86e7T^{2}
37 1−1.31e4T+6.93e7T2 1 - 1.31e4T + 6.93e7T^{2}
41 1+1.70e4T+1.15e8T2 1 + 1.70e4T + 1.15e8T^{2}
43 1−8.93e3T+1.47e8T2 1 - 8.93e3T + 1.47e8T^{2}
47 1+1.57e4T+2.29e8T2 1 + 1.57e4T + 2.29e8T^{2}
53 1+4.03e4T+4.18e8T2 1 + 4.03e4T + 4.18e8T^{2}
59 1+4.75e4T+7.14e8T2 1 + 4.75e4T + 7.14e8T^{2}
61 1+3.02e4T+8.44e8T2 1 + 3.02e4T + 8.44e8T^{2}
67 1+3.87e4T+1.35e9T2 1 + 3.87e4T + 1.35e9T^{2}
71 1+1.05e4T+1.80e9T2 1 + 1.05e4T + 1.80e9T^{2}
73 1+1.58e3T+2.07e9T2 1 + 1.58e3T + 2.07e9T^{2}
79 1+6.19e3T+3.07e9T2 1 + 6.19e3T + 3.07e9T^{2}
83 1+3.78e4T+3.93e9T2 1 + 3.78e4T + 3.93e9T^{2}
89 1−4.91e4T+5.58e9T2 1 - 4.91e4T + 5.58e9T^{2}
97 1−1.56e4T+8.58e9T2 1 - 1.56e4T + 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

−9.493437377566180999564146043846, −8.765670026982283328815422666282, −8.091084382420908790824892733140, −7.33961416325794839205208621935, −6.32779923722187225633952688426, −4.52782093436346387701863180333, −4.42700831901146259938861051574, −3.14965351235742196323327859547, −1.99113021213989685235262906358, −1.52453463465547112685543993256, 1.52453463465547112685543993256, 1.99113021213989685235262906358, 3.14965351235742196323327859547, 4.42700831901146259938861051574, 4.52782093436346387701863180333, 6.32779923722187225633952688426, 7.33961416325794839205208621935, 8.091084382420908790824892733140, 8.765670026982283328815422666282, 9.493437377566180999564146043846

Graph of the ZZ-function along the critical line