↑

Properties

Label 2-624-1.1-c5-0-43
Degree 22
Conductor 624624
Sign −1-1
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 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s − 61.4·5-s + 46.3·7-s + 81·9-s − 16.2·11-s − 169·13-s − 552.·15-s + 659.·17-s − 17.7·19-s + 416.·21-s + 2.14e3·23-s + 646.·25-s + 729·27-s − 3.17e3·29-s − 4.03e3·31-s − 146.·33-s − 2.84e3·35-s + 1.42e4·37-s − 1.52e3·39-s − 1.78e4·41-s + 3.55e3·43-s − 4.97e3·45-s + 2.52e4·47-s − 1.46e4·49-s + 5.93e3·51-s − 5.30e3·53-s + 1.00e3·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.09·5-s + 0.357·7-s + 0.333·9-s − 0.0406·11-s − 0.277·13-s − 0.634·15-s + 0.553·17-s − 0.0112·19-s + 0.206·21-s + 0.843·23-s + 0.207·25-s + 0.192·27-s − 0.701·29-s − 0.753·31-s − 0.0234·33-s − 0.392·35-s + 1.71·37-s − 0.160·39-s − 1.66·41-s + 0.293·43-s − 0.366·45-s + 1.66·47-s − 0.872·49-s + 0.319·51-s − 0.259·53-s + 0.0446·55-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: −1-1
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: 11
Selberg data: (2, 624, ( :5/2), −1)(2,\ 624,\ (\ :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)
bad2 1 1
3 1−9T 1 - 9T
13 1+169T 1 + 169T
good5 1+61.4T+3.12e3T2 1 + 61.4T + 3.12e3T^{2}
7 1−46.3T+1.68e4T2 1 - 46.3T + 1.68e4T^{2}
11 1+16.2T+1.61e5T2 1 + 16.2T + 1.61e5T^{2}
17 1−659.T+1.41e6T2 1 - 659.T + 1.41e6T^{2}
19 1+17.7T+2.47e6T2 1 + 17.7T + 2.47e6T^{2}
23 1−2.14e3T+6.43e6T2 1 - 2.14e3T + 6.43e6T^{2}
29 1+3.17e3T+2.05e7T2 1 + 3.17e3T + 2.05e7T^{2}
31 1+4.03e3T+2.86e7T2 1 + 4.03e3T + 2.86e7T^{2}
37 1−1.42e4T+6.93e7T2 1 - 1.42e4T + 6.93e7T^{2}
41 1+1.78e4T+1.15e8T2 1 + 1.78e4T + 1.15e8T^{2}
43 1−3.55e3T+1.47e8T2 1 - 3.55e3T + 1.47e8T^{2}
47 1−2.52e4T+2.29e8T2 1 - 2.52e4T + 2.29e8T^{2}
53 1+5.30e3T+4.18e8T2 1 + 5.30e3T + 4.18e8T^{2}
59 1−3.65e4T+7.14e8T2 1 - 3.65e4T + 7.14e8T^{2}
61 1+6.51e3T+8.44e8T2 1 + 6.51e3T + 8.44e8T^{2}
67 1+4.11e4T+1.35e9T2 1 + 4.11e4T + 1.35e9T^{2}
71 1+2.09e4T+1.80e9T2 1 + 2.09e4T + 1.80e9T^{2}
73 1+7.10e4T+2.07e9T2 1 + 7.10e4T + 2.07e9T^{2}
79 1+4.91e4T+3.07e9T2 1 + 4.91e4T + 3.07e9T^{2}
83 1+9.99e4T+3.93e9T2 1 + 9.99e4T + 3.93e9T^{2}
89 1−4.72e4T+5.58e9T2 1 - 4.72e4T + 5.58e9T^{2}
97 1+3.97e4T+8.58e9T2 1 + 3.97e4T + 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

−9.270342274518542379589465453941, −8.431946541442639061809595164243, −7.65000389226835024268417483466, −7.09533756395803001326677995271, −5.66752325415361096522758438260, −4.54793615613510042432072461045, −3.71161525396971685437646984392, −2.72019310846678036340228897972, −1.34300963898326966150701566625, 0, 1.34300963898326966150701566625, 2.72019310846678036340228897972, 3.71161525396971685437646984392, 4.54793615613510042432072461045, 5.66752325415361096522758438260, 7.09533756395803001326677995271, 7.65000389226835024268417483466, 8.431946541442639061809595164243, 9.270342274518542379589465453941

Graph of the ZZ-function along the critical line