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Properties

Label 2-252-1.1-c3-0-5
Degree 22
Conductor 252252
Sign −1-1
Analytic cond. 14.868414.8684
Root an. cond. 3.855963.85596
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s + 7·7-s − 36·11-s + 62·13-s − 114·17-s − 76·19-s + 24·23-s − 89·25-s − 54·29-s − 112·31-s − 42·35-s − 178·37-s − 378·41-s − 172·43-s + 192·47-s + 49·49-s + 402·53-s + 216·55-s − 396·59-s + 254·61-s − 372·65-s − 1.01e3·67-s − 840·71-s + 890·73-s − 252·77-s + 80·79-s + 108·83-s + ⋯
L(s)  = 1  − 0.536·5-s + 0.377·7-s − 0.986·11-s + 1.32·13-s − 1.62·17-s − 0.917·19-s + 0.217·23-s − 0.711·25-s − 0.345·29-s − 0.648·31-s − 0.202·35-s − 0.790·37-s − 1.43·41-s − 0.609·43-s + 0.595·47-s + 1/7·49-s + 1.04·53-s + 0.529·55-s − 0.873·59-s + 0.533·61-s − 0.709·65-s − 1.84·67-s − 1.40·71-s + 1.42·73-s − 0.372·77-s + 0.113·79-s + 0.142·83-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, 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: 14.868414.8684
Root analytic conductor: 3.855963.85596
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 252, ( :3/2), −1)(2,\ 252,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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 1
7 1−pT 1 - p T
good5 1+6T+p3T2 1 + 6 T + p^{3} T^{2}
11 1+36T+p3T2 1 + 36 T + p^{3} T^{2}
13 1−62T+p3T2 1 - 62 T + p^{3} T^{2}
17 1+114T+p3T2 1 + 114 T + p^{3} T^{2}
19 1+4pT+p3T2 1 + 4 p T + p^{3} T^{2}
23 1−24T+p3T2 1 - 24 T + p^{3} T^{2}
29 1+54T+p3T2 1 + 54 T + p^{3} T^{2}
31 1+112T+p3T2 1 + 112 T + p^{3} T^{2}
37 1+178T+p3T2 1 + 178 T + p^{3} T^{2}
41 1+378T+p3T2 1 + 378 T + p^{3} T^{2}
43 1+4pT+p3T2 1 + 4 p T + p^{3} T^{2}
47 1−192T+p3T2 1 - 192 T + p^{3} T^{2}
53 1−402T+p3T2 1 - 402 T + p^{3} T^{2}
59 1+396T+p3T2 1 + 396 T + p^{3} T^{2}
61 1−254T+p3T2 1 - 254 T + p^{3} T^{2}
67 1+1012T+p3T2 1 + 1012 T + p^{3} T^{2}
71 1+840T+p3T2 1 + 840 T + p^{3} T^{2}
73 1−890T+p3T2 1 - 890 T + p^{3} T^{2}
79 1−80T+p3T2 1 - 80 T + p^{3} T^{2}
83 1−108T+p3T2 1 - 108 T + p^{3} T^{2}
89 1−1638T+p3T2 1 - 1638 T + p^{3} T^{2}
97 1−1010T+p3T2 1 - 1010 T + p^{3} 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

−11.02851742986436830150646304293, −10.48109944032639874377566182827, −8.909415316301294582046174060833, −8.306413665781800601275156479431, −7.18515319544684163584848087811, −6.03509074832949923859509344246, −4.74695254805227976499825147575, −3.63460833012397145434805269090, −1.99987871287688285887259941607, 0, 1.99987871287688285887259941607, 3.63460833012397145434805269090, 4.74695254805227976499825147575, 6.03509074832949923859509344246, 7.18515319544684163584848087811, 8.306413665781800601275156479431, 8.909415316301294582046174060833, 10.48109944032639874377566182827, 11.02851742986436830150646304293

Graph of the ZZ-function along the critical line