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Properties

Label 2-252-1.1-c5-0-5
Degree 22
Conductor 252252
Sign 11
Analytic cond. 40.416740.4167
Root an. cond. 6.357416.35741
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 96·5-s + 49·7-s + 720·11-s + 572·13-s − 1.25e3·17-s − 94·19-s − 96·23-s + 6.09e3·25-s + 4.37e3·29-s − 6.24e3·31-s + 4.70e3·35-s − 1.07e4·37-s − 1.20e4·41-s − 9.16e3·43-s + 2.58e4·47-s + 2.40e3·49-s − 1.01e3·53-s + 6.91e4·55-s − 1.24e3·59-s + 7.59e3·61-s + 5.49e4·65-s + 4.11e4·67-s + 3.76e4·71-s − 1.34e4·73-s + 3.52e4·77-s + 6.24e3·79-s + 2.52e4·83-s + ⋯
L(s)  = 1  + 1.71·5-s + 0.377·7-s + 1.79·11-s + 0.938·13-s − 1.05·17-s − 0.0597·19-s − 0.0378·23-s + 1.94·25-s + 0.965·29-s − 1.16·31-s + 0.649·35-s − 1.29·37-s − 1.11·41-s − 0.755·43-s + 1.70·47-s + 1/7·49-s − 0.0495·53-s + 3.08·55-s − 0.0464·59-s + 0.261·61-s + 1.61·65-s + 1.11·67-s + 0.885·71-s − 0.295·73-s + 0.678·77-s + 0.112·79-s + 0.402·83-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: 11
Analytic conductor: 40.416740.4167
Root analytic conductor: 6.357416.35741
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 252, ( :5/2), 1)(2,\ 252,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 3.4172322033.417232203
L(12)L(\frac12) ≈\approx 3.4172322033.417232203
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 1
7 1−p2T 1 - p^{2} T
good5 1−96T+p5T2 1 - 96 T + p^{5} T^{2}
11 1−720T+p5T2 1 - 720 T + p^{5} T^{2}
13 1−44pT+p5T2 1 - 44 p T + p^{5} T^{2}
17 1+1254T+p5T2 1 + 1254 T + p^{5} T^{2}
19 1+94T+p5T2 1 + 94 T + p^{5} T^{2}
23 1+96T+p5T2 1 + 96 T + p^{5} T^{2}
29 1−4374T+p5T2 1 - 4374 T + p^{5} T^{2}
31 1+6244T+p5T2 1 + 6244 T + p^{5} T^{2}
37 1+10798T+p5T2 1 + 10798 T + p^{5} T^{2}
41 1+12006T+p5T2 1 + 12006 T + p^{5} T^{2}
43 1+9160T+p5T2 1 + 9160 T + p^{5} T^{2}
47 1−25836T+p5T2 1 - 25836 T + p^{5} T^{2}
53 1+1014T+p5T2 1 + 1014 T + p^{5} T^{2}
59 1+1242T+p5T2 1 + 1242 T + p^{5} T^{2}
61 1−7592T+p5T2 1 - 7592 T + p^{5} T^{2}
67 1−41132T+p5T2 1 - 41132 T + p^{5} T^{2}
71 1−37632T+p5T2 1 - 37632 T + p^{5} T^{2}
73 1+13438T+p5T2 1 + 13438 T + p^{5} T^{2}
79 1−6248T+p5T2 1 - 6248 T + p^{5} T^{2}
83 1−25254T+p5T2 1 - 25254 T + p^{5} T^{2}
89 1−45126T+p5T2 1 - 45126 T + p^{5} T^{2}
97 1−107222T+p5T2 1 - 107222 T + p^{5} 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

−11.07997769004327497873864175886, −10.19905114445934005235328822324, −9.113586937523096138048897295155, −8.708036719053210425503338138769, −6.80923408949690372420803922762, −6.25776464124503790928344988874, −5.13878585087194986800969731221, −3.76771266103547017919565726086, −2.07592691154418043550096265874, −1.24359954126291028944787993055, 1.24359954126291028944787993055, 2.07592691154418043550096265874, 3.76771266103547017919565726086, 5.13878585087194986800969731221, 6.25776464124503790928344988874, 6.80923408949690372420803922762, 8.708036719053210425503338138769, 9.113586937523096138048897295155, 10.19905114445934005235328822324, 11.07997769004327497873864175886

Graph of the ZZ-function along the critical line