↑

Properties

Label 2-912-1.1-c5-0-8
Degree 22
Conductor 912912
Sign 11
Analytic cond. 146.270146.270
Root an. cond. 12.094212.0942
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 9·3-s − 69.8·5-s + 147.·7-s + 81·9-s − 294.·11-s − 1.11e3·13-s + 628.·15-s + 1.89e3·17-s − 361·19-s − 1.33e3·21-s − 828.·23-s + 1.75e3·25-s − 729·27-s + 1.43e3·29-s − 1.02e4·31-s + 2.65e3·33-s − 1.03e4·35-s + 1.42e4·37-s + 1.00e4·39-s − 1.37e4·41-s + 701.·43-s − 5.65e3·45-s + 3.56e3·47-s + 5.07e3·49-s − 1.70e4·51-s − 2.94e4·53-s + 2.05e4·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.24·5-s + 1.14·7-s + 0.333·9-s − 0.733·11-s − 1.82·13-s + 0.721·15-s + 1.59·17-s − 0.229·19-s − 0.658·21-s − 0.326·23-s + 0.560·25-s − 0.192·27-s + 0.317·29-s − 1.92·31-s + 0.423·33-s − 1.42·35-s + 1.71·37-s + 1.05·39-s − 1.27·41-s + 0.0578·43-s − 0.416·45-s + 0.235·47-s + 0.301·49-s − 0.918·51-s − 1.43·53-s + 0.916·55-s + ⋯

Functional equation

Λ(s)=(912s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(912s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 912912    =    24⋅3⋅192^{4} \cdot 3 \cdot 19
Sign: 11
Analytic conductor: 146.270146.270
Root analytic conductor: 12.094212.0942
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 912, ( :5/2), 1)(2,\ 912,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.71854967940.7185496794
L(12)L(\frac12) ≈\approx 0.71854967940.7185496794
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 1
3 1+9T 1 + 9T
19 1+361T 1 + 361T
good5 1+69.8T+3.12e3T2 1 + 69.8T + 3.12e3T^{2}
7 1−147.T+1.68e4T2 1 - 147.T + 1.68e4T^{2}
11 1+294.T+1.61e5T2 1 + 294.T + 1.61e5T^{2}
13 1+1.11e3T+3.71e5T2 1 + 1.11e3T + 3.71e5T^{2}
17 1−1.89e3T+1.41e6T2 1 - 1.89e3T + 1.41e6T^{2}
23 1+828.T+6.43e6T2 1 + 828.T + 6.43e6T^{2}
29 1−1.43e3T+2.05e7T2 1 - 1.43e3T + 2.05e7T^{2}
31 1+1.02e4T+2.86e7T2 1 + 1.02e4T + 2.86e7T^{2}
37 1−1.42e4T+6.93e7T2 1 - 1.42e4T + 6.93e7T^{2}
41 1+1.37e4T+1.15e8T2 1 + 1.37e4T + 1.15e8T^{2}
43 1−701.T+1.47e8T2 1 - 701.T + 1.47e8T^{2}
47 1−3.56e3T+2.29e8T2 1 - 3.56e3T + 2.29e8T^{2}
53 1+2.94e4T+4.18e8T2 1 + 2.94e4T + 4.18e8T^{2}
59 1+3.06e4T+7.14e8T2 1 + 3.06e4T + 7.14e8T^{2}
61 1−4.03e4T+8.44e8T2 1 - 4.03e4T + 8.44e8T^{2}
67 1+6.37e4T+1.35e9T2 1 + 6.37e4T + 1.35e9T^{2}
71 1−6.08e4T+1.80e9T2 1 - 6.08e4T + 1.80e9T^{2}
73 1+2.59e4T+2.07e9T2 1 + 2.59e4T + 2.07e9T^{2}
79 1+3.92e4T+3.07e9T2 1 + 3.92e4T + 3.07e9T^{2}
83 1+9.11e4T+3.93e9T2 1 + 9.11e4T + 3.93e9T^{2}
89 1−1.73e4T+5.58e9T2 1 - 1.73e4T + 5.58e9T^{2}
97 1−1.25e5T+8.58e9T2 1 - 1.25e5T + 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

−9.476686937765708534219278422262, −8.051539890774414030601048727174, −7.78192688407917461265393453668, −7.11030405443795111187365034674, −5.61715324944171958902494650470, −4.95543477296360769315175367153, −4.21936332544465385039572758238, −3.00866459229885942021109290571, −1.71355071877995380878807591149, −0.38526182465565005230588022823, 0.38526182465565005230588022823, 1.71355071877995380878807591149, 3.00866459229885942021109290571, 4.21936332544465385039572758238, 4.95543477296360769315175367153, 5.61715324944171958902494650470, 7.11030405443795111187365034674, 7.78192688407917461265393453668, 8.051539890774414030601048727174, 9.476686937765708534219278422262

Graph of the ZZ-function along the critical line