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Properties

Label 2-252-1.1-c11-0-24
Degree 22
Conductor 252252
Sign −1-1
Analytic cond. 193.622193.622
Root an. cond. 13.914813.9148
Motivic weight 1111
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.13e3·5-s + 1.68e4·7-s + 7.04e5·11-s + 9.52e5·13-s − 5.10e6·17-s − 1.39e7·19-s + 1.89e7·23-s − 4.42e7·25-s − 1.48e8·29-s − 1.59e8·31-s + 3.57e7·35-s − 8.25e7·37-s + 7.29e8·41-s + 1.18e9·43-s − 2.86e8·47-s + 2.82e8·49-s − 3.85e9·53-s + 1.49e9·55-s − 5.28e9·59-s − 8.15e9·61-s + 2.02e9·65-s + 9.25e9·67-s − 2.00e10·71-s + 2.38e10·73-s + 1.18e10·77-s + 3.51e9·79-s + 2.14e10·83-s + ⋯
L(s)  = 1  + 0.304·5-s + 0.377·7-s + 1.31·11-s + 0.711·13-s − 0.872·17-s − 1.28·19-s + 0.613·23-s − 0.907·25-s − 1.34·29-s − 0.998·31-s + 0.115·35-s − 0.195·37-s + 0.983·41-s + 1.22·43-s − 0.181·47-s + 1/7·49-s − 1.26·53-s + 0.401·55-s − 0.963·59-s − 1.23·61-s + 0.216·65-s + 0.837·67-s − 1.31·71-s + 1.34·73-s + 0.498·77-s + 0.128·79-s + 0.599·83-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=(− Λ(12−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+11/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: −1-1
Analytic conductor: 193.622193.622
Root analytic conductor: 13.914813.9148
Motivic weight: 1111
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 252, ( :11/2), −1)(2,\ 252,\ (\ :11/2),\ -1)

Particular Values

L(6)L(6) == 00
L(12)L(\frac12) == 00
L(132)L(\frac{13}{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 1
7 1−p5T 1 - p^{5} T
good5 1−426pT+p11T2 1 - 426 p T + p^{11} T^{2}
11 1−704196T+p11T2 1 - 704196 T + p^{11} T^{2}
13 1−952286T+p11T2 1 - 952286 T + p^{11} T^{2}
17 1+5105514T+p11T2 1 + 5105514 T + p^{11} T^{2}
19 1+13905148T+p11T2 1 + 13905148 T + p^{11} T^{2}
23 1−18945408T+p11T2 1 - 18945408 T + p^{11} T^{2}
29 1+148937094T+p11T2 1 + 148937094 T + p^{11} T^{2}
31 1+159226144T+p11T2 1 + 159226144 T + p^{11} T^{2}
37 1+82548178T+p11T2 1 + 82548178 T + p^{11} T^{2}
41 1−729417150T+p11T2 1 - 729417150 T + p^{11} T^{2}
43 1−1185139028T+p11T2 1 - 1185139028 T + p^{11} T^{2}
47 1+286058928T+p11T2 1 + 286058928 T + p^{11} T^{2}
53 1+3853540014T+p11T2 1 + 3853540014 T + p^{11} T^{2}
59 1+5288267196T+p11T2 1 + 5288267196 T + p^{11} T^{2}
61 1+8156327602T+p11T2 1 + 8156327602 T + p^{11} T^{2}
67 1−9250048316T+p11T2 1 - 9250048316 T + p^{11} T^{2}
71 1+20051655792T+p11T2 1 + 20051655792 T + p^{11} T^{2}
73 1−23853193802T+p11T2 1 - 23853193802 T + p^{11} T^{2}
79 1−3513675584T+p11T2 1 - 3513675584 T + p^{11} T^{2}
83 1−21497352012T+p11T2 1 - 21497352012 T + p^{11} T^{2}
89 1+66839945634T+p11T2 1 + 66839945634 T + p^{11} T^{2}
97 1−146492724002T+p11T2 1 - 146492724002 T + p^{11} T^{2}
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   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

−9.352246845069877287640552881366, −8.910672125671937576845578933371, −7.69504301847692952963252882426, −6.56234780018984140500060394916, −5.83206671855053102816213312379, −4.45168562517235053739785194550, −3.67568081391644812208881531697, −2.14347869546164577084595817277, −1.36119925682415161896920069780, 0, 1.36119925682415161896920069780, 2.14347869546164577084595817277, 3.67568081391644812208881531697, 4.45168562517235053739785194550, 5.83206671855053102816213312379, 6.56234780018984140500060394916, 7.69504301847692952963252882426, 8.910672125671937576845578933371, 9.352246845069877287640552881366

Graph of the ZZ-function along the critical line