↑

Properties

Label 2-325-1.1-c5-0-48
Degree 22
Conductor 325325
Sign −1-1
Analytic cond. 52.124752.1247
Root an. cond. 7.219747.21974
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.75·2-s − 23.9·3-s − 24.4·4-s − 66.0·6-s − 85.7·7-s − 155.·8-s + 331.·9-s + 431.·11-s + 584.·12-s − 169·13-s − 236.·14-s + 352.·16-s + 438.·17-s + 912.·18-s + 1.61e3·19-s + 2.05e3·21-s + 1.18e3·22-s − 2.17e3·23-s + 3.72e3·24-s − 465.·26-s − 2.11e3·27-s + 2.09e3·28-s + 8.28e3·29-s − 2.74e3·31-s + 5.94e3·32-s − 1.03e4·33-s + 1.20e3·34-s + ⋯
L(s)  = 1  + 0.487·2-s − 1.53·3-s − 0.762·4-s − 0.749·6-s − 0.661·7-s − 0.858·8-s + 1.36·9-s + 1.07·11-s + 1.17·12-s − 0.277·13-s − 0.322·14-s + 0.343·16-s + 0.368·17-s + 0.664·18-s + 1.02·19-s + 1.01·21-s + 0.523·22-s − 0.856·23-s + 1.32·24-s − 0.135·26-s − 0.557·27-s + 0.504·28-s + 1.83·29-s − 0.513·31-s + 1.02·32-s − 1.65·33-s + 0.179·34-s + ⋯

Functional equation

Λ(s)=(325s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(325s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 325325    =    52⋅135^{2} \cdot 13
Sign: −1-1
Analytic conductor: 52.124752.1247
Root analytic conductor: 7.219747.21974
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 325, ( :5/2), −1)(2,\ 325,\ (\ :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)
bad5 1 1
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}
show more
show less
   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

−10.21350646656407414254254783013, −9.707675641774025140304302535021, −8.513772443252950137087218591348, −6.95908850932545750520435339210, −6.16036106965174051724777938539, −5.34634751750776506845171525647, −4.43428335238129209060214704046, −3.33289960411406634424279978995, −1.07708091498980695236644692499, 0, 1.07708091498980695236644692499, 3.33289960411406634424279978995, 4.43428335238129209060214704046, 5.34634751750776506845171525647, 6.16036106965174051724777938539, 6.95908850932545750520435339210, 8.513772443252950137087218591348, 9.707675641774025140304302535021, 10.21350646656407414254254783013

Graph of the ZZ-function along the critical line