↑

Properties

Label 2-624-1.1-c5-0-35
Degree 22
Conductor 624624
Sign 11
Analytic cond. 100.079100.079
Root an. cond. 10.003910.0039
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 + 108.·5-s + 213.·7-s + 81·9-s − 401.·11-s − 169·13-s + 973.·15-s − 1.70e3·17-s + 373.·19-s + 1.92e3·21-s + 2.35e3·23-s + 8.56e3·25-s + 729·27-s − 2.93e3·29-s + 2.06e3·31-s − 3.60e3·33-s + 2.30e4·35-s + 1.39e4·37-s − 1.52e3·39-s + 3.62e3·41-s + 1.81e4·43-s + 8.75e3·45-s − 1.18e4·47-s + 2.87e4·49-s − 1.53e4·51-s − 1.34e4·53-s − 4.33e4·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.93·5-s + 1.64·7-s + 0.333·9-s − 0.999·11-s − 0.277·13-s + 1.11·15-s − 1.43·17-s + 0.237·19-s + 0.950·21-s + 0.926·23-s + 2.74·25-s + 0.192·27-s − 0.647·29-s + 0.385·31-s − 0.576·33-s + 3.18·35-s + 1.67·37-s − 0.160·39-s + 0.337·41-s + 1.49·43-s + 0.644·45-s − 0.780·47-s + 1.71·49-s − 0.827·51-s − 0.656·53-s − 1.93·55-s + ⋯

Functional equation

Λ(s)=(624s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(624s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 624624    =    24⋅3⋅132^{4} \cdot 3 \cdot 13
Sign: 11
Analytic conductor: 100.079100.079
Root analytic conductor: 10.003910.0039
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 624, ( :5/2), 1)(2,\ 624,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 5.0158354635.015835463
L(12)L(\frac12) ≈\approx 5.0158354635.015835463
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
13 1+169T 1 + 169T
good5 1−108.T+3.12e3T2 1 - 108.T + 3.12e3T^{2}
7 1−213.T+1.68e4T2 1 - 213.T + 1.68e4T^{2}
11 1+401.T+1.61e5T2 1 + 401.T + 1.61e5T^{2}
17 1+1.70e3T+1.41e6T2 1 + 1.70e3T + 1.41e6T^{2}
19 1−373.T+2.47e6T2 1 - 373.T + 2.47e6T^{2}
23 1−2.35e3T+6.43e6T2 1 - 2.35e3T + 6.43e6T^{2}
29 1+2.93e3T+2.05e7T2 1 + 2.93e3T + 2.05e7T^{2}
31 1−2.06e3T+2.86e7T2 1 - 2.06e3T + 2.86e7T^{2}
37 1−1.39e4T+6.93e7T2 1 - 1.39e4T + 6.93e7T^{2}
41 1−3.62e3T+1.15e8T2 1 - 3.62e3T + 1.15e8T^{2}
43 1−1.81e4T+1.47e8T2 1 - 1.81e4T + 1.47e8T^{2}
47 1+1.18e4T+2.29e8T2 1 + 1.18e4T + 2.29e8T^{2}
53 1+1.34e4T+4.18e8T2 1 + 1.34e4T + 4.18e8T^{2}
59 1−2.98e4T+7.14e8T2 1 - 2.98e4T + 7.14e8T^{2}
61 1+2.62e4T+8.44e8T2 1 + 2.62e4T + 8.44e8T^{2}
67 1−5.03e3T+1.35e9T2 1 - 5.03e3T + 1.35e9T^{2}
71 1+1.49e4T+1.80e9T2 1 + 1.49e4T + 1.80e9T^{2}
73 1+3.41e4T+2.07e9T2 1 + 3.41e4T + 2.07e9T^{2}
79 1−1.06e5T+3.07e9T2 1 - 1.06e5T + 3.07e9T^{2}
83 1−9.81e4T+3.93e9T2 1 - 9.81e4T + 3.93e9T^{2}
89 1+3.22e4T+5.58e9T2 1 + 3.22e4T + 5.58e9T^{2}
97 1+1.24e5T+8.58e9T2 1 + 1.24e5T + 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

−9.678898849685064217966673390695, −9.061704465305056461415882129486, −8.189248379587232770798656981448, −7.27456146439643197788361532711, −6.12491152672947748266161869564, −5.16606387490170922762089770562, −4.57653853870386938243487044266, −2.61482769082547122413051815636, −2.14040649820841515340816500108, −1.13800346869777333813957498659, 1.13800346869777333813957498659, 2.14040649820841515340816500108, 2.61482769082547122413051815636, 4.57653853870386938243487044266, 5.16606387490170922762089770562, 6.12491152672947748266161869564, 7.27456146439643197788361532711, 8.189248379587232770798656981448, 9.061704465305056461415882129486, 9.678898849685064217966673390695

Graph of the ZZ-function along the critical line