↑

Properties

Label 2-2496-1.1-c3-0-38
Degree 22
Conductor 24962496
Sign 11
Analytic cond. 147.268147.268
Root an. cond. 12.135412.1354
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 12.8·5-s + 24.8·7-s + 9·9-s − 19.6·11-s + 13·13-s − 38.4·15-s − 63.6·17-s + 0.832·19-s + 74.4·21-s + 119.·23-s + 39.6·25-s + 27·27-s + 6·29-s + 185.·31-s − 58.9·33-s − 318.·35-s − 143.·37-s + 39·39-s − 117.·41-s − 67.6·43-s − 115.·45-s + 476.·47-s + 273.·49-s − 190.·51-s − 59.3·53-s + 252.·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.14·5-s + 1.34·7-s + 0.333·9-s − 0.539·11-s + 0.277·13-s − 0.662·15-s − 0.908·17-s + 0.0100·19-s + 0.774·21-s + 1.08·23-s + 0.317·25-s + 0.192·27-s + 0.0384·29-s + 1.07·31-s − 0.311·33-s − 1.53·35-s − 0.638·37-s + 0.160·39-s − 0.446·41-s − 0.240·43-s − 0.382·45-s + 1.47·47-s + 0.797·49-s − 0.524·51-s − 0.153·53-s + 0.618·55-s + ⋯

Functional equation

Λ(s)=(2496s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(2496s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24962496    =    26⋅3⋅132^{6} \cdot 3 \cdot 13
Sign: 11
Analytic conductor: 147.268147.268
Root analytic conductor: 12.135412.1354
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2496, ( :3/2), 1)(2,\ 2496,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.4467302492.446730249
L(12)L(\frac12) ≈\approx 2.4467302492.446730249
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)
bad2 1 1
3 1−3T 1 - 3T
13 1−13T 1 - 13T
good5 1+12.8T+125T2 1 + 12.8T + 125T^{2}
7 1−24.8T+343T2 1 - 24.8T + 343T^{2}
11 1+19.6T+1.33e3T2 1 + 19.6T + 1.33e3T^{2}
17 1+63.6T+4.91e3T2 1 + 63.6T + 4.91e3T^{2}
19 1−0.832T+6.85e3T2 1 - 0.832T + 6.85e3T^{2}
23 1−119.T+1.21e4T2 1 - 119.T + 1.21e4T^{2}
29 1−6T+2.43e4T2 1 - 6T + 2.43e4T^{2}
31 1−185.T+2.97e4T2 1 - 185.T + 2.97e4T^{2}
37 1+143.T+5.06e4T2 1 + 143.T + 5.06e4T^{2}
41 1+117.T+6.89e4T2 1 + 117.T + 6.89e4T^{2}
43 1+67.6T+7.95e4T2 1 + 67.6T + 7.95e4T^{2}
47 1−476.T+1.03e5T2 1 - 476.T + 1.03e5T^{2}
53 1+59.3T+1.48e5T2 1 + 59.3T + 1.48e5T^{2}
59 1+78T+2.05e5T2 1 + 78T + 2.05e5T^{2}
61 1+609.T+2.26e5T2 1 + 609.T + 2.26e5T^{2}
67 1−654.T+3.00e5T2 1 - 654.T + 3.00e5T^{2}
71 1+390.T+3.57e5T2 1 + 390.T + 3.57e5T^{2}
73 1+84.3T+3.89e5T2 1 + 84.3T + 3.89e5T^{2}
79 1−1.17e3T+4.93e5T2 1 - 1.17e3T + 4.93e5T^{2}
83 1−430.T+5.71e5T2 1 - 430.T + 5.71e5T^{2}
89 1+1.34e3T+7.04e5T2 1 + 1.34e3T + 7.04e5T^{2}
97 1−802.T+9.12e5T2 1 - 802.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

−8.580246664667339829285116039000, −7.78734772325405880652713571009, −7.40208562310227222594430794238, −6.41541411065145478510039068912, −5.11573464981310914896124032192, −4.58776790855531601925819629576, −3.79069705725521768611067178115, −2.81809388124341308765377314636, −1.81569171989624458601438818509, −0.67560082620932839629182886096, 0.67560082620932839629182886096, 1.81569171989624458601438818509, 2.81809388124341308765377314636, 3.79069705725521768611067178115, 4.58776790855531601925819629576, 5.11573464981310914896124032192, 6.41541411065145478510039068912, 7.40208562310227222594430794238, 7.78734772325405880652713571009, 8.580246664667339829285116039000

Graph of the ZZ-function along the critical line