↑

Properties

Label 2-77-1.1-c3-0-15
Degree 22
Conductor 7777
Sign −1-1
Analytic cond. 4.543144.54314
Root an. cond. 2.131462.13146
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.76·2-s − 7.36·3-s + 6.16·4-s − 15.4·5-s − 27.7·6-s − 7·7-s − 6.90·8-s + 27.2·9-s − 58.3·10-s + 11·11-s − 45.3·12-s + 49.0·13-s − 26.3·14-s + 114.·15-s − 75.3·16-s − 34.1·17-s + 102.·18-s − 144.·19-s − 95.5·20-s + 51.5·21-s + 41.4·22-s + 118.·23-s + 50.8·24-s + 115.·25-s + 184.·26-s − 1.63·27-s − 43.1·28-s + ⋯
L(s)  = 1  + 1.33·2-s − 1.41·3-s + 0.770·4-s − 1.38·5-s − 1.88·6-s − 0.377·7-s − 0.305·8-s + 1.00·9-s − 1.84·10-s + 0.301·11-s − 1.09·12-s + 1.04·13-s − 0.502·14-s + 1.96·15-s − 1.17·16-s − 0.486·17-s + 1.34·18-s − 1.74·19-s − 1.06·20-s + 0.535·21-s + 0.401·22-s + 1.07·23-s + 0.432·24-s + 0.920·25-s + 1.39·26-s − 0.0116·27-s − 0.291·28-s + ⋯

Functional equation

Λ(s)=(77s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(77s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7777    =    7⋅117 \cdot 11
Sign: −1-1
Analytic conductor: 4.543144.54314
Root analytic conductor: 2.131462.13146
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 77, ( :3/2), −1)(2,\ 77,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad7 1+7T 1 + 7T
11 1−11T 1 - 11T
good2 1−3.76T+8T2 1 - 3.76T + 8T^{2}
3 1+7.36T+27T2 1 + 7.36T + 27T^{2}
5 1+15.4T+125T2 1 + 15.4T + 125T^{2}
13 1−49.0T+2.19e3T2 1 - 49.0T + 2.19e3T^{2}
17 1+34.1T+4.91e3T2 1 + 34.1T + 4.91e3T^{2}
19 1+144.T+6.85e3T2 1 + 144.T + 6.85e3T^{2}
23 1−118.T+1.21e4T2 1 - 118.T + 1.21e4T^{2}
29 1+63.6T+2.43e4T2 1 + 63.6T + 2.43e4T^{2}
31 1+212.T+2.97e4T2 1 + 212.T + 2.97e4T^{2}
37 1+200.T+5.06e4T2 1 + 200.T + 5.06e4T^{2}
41 1−451.T+6.89e4T2 1 - 451.T + 6.89e4T^{2}
43 1+130.T+7.95e4T2 1 + 130.T + 7.95e4T^{2}
47 1+176.T+1.03e5T2 1 + 176.T + 1.03e5T^{2}
53 1+629.T+1.48e5T2 1 + 629.T + 1.48e5T^{2}
59 1+86.9T+2.05e5T2 1 + 86.9T + 2.05e5T^{2}
61 1−644.T+2.26e5T2 1 - 644.T + 2.26e5T^{2}
67 1+400.T+3.00e5T2 1 + 400.T + 3.00e5T^{2}
71 1−507.T+3.57e5T2 1 - 507.T + 3.57e5T^{2}
73 1−176.T+3.89e5T2 1 - 176.T + 3.89e5T^{2}
79 1+701.T+4.93e5T2 1 + 701.T + 4.93e5T^{2}
83 1+1.25e3T+5.71e5T2 1 + 1.25e3T + 5.71e5T^{2}
89 1−788.T+7.04e5T2 1 - 788.T + 7.04e5T^{2}
97 1−185.T+9.12e5T2 1 - 185.T + 9.12e5T^{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

−12.98290294102715277677537190096, −12.50885535221936243262932407851, −11.31106202974706013077606134142, −11.00490809111876813323187937030, −8.758606309476125421947229676998, −6.89947703437141300004680603572, −5.98983998150940296546754602579, −4.63810192447921754194094079760, −3.67358803512856225534933034657, 0, 3.67358803512856225534933034657, 4.63810192447921754194094079760, 5.98983998150940296546754602579, 6.89947703437141300004680603572, 8.758606309476125421947229676998, 11.00490809111876813323187937030, 11.31106202974706013077606134142, 12.50885535221936243262932407851, 12.98290294102715277677537190096

Graph of the ZZ-function along the critical line