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Properties

Label 4-162e2-1.1-c3e2-0-5
Degree 44
Conductor 2624426244
Sign 11
Analytic cond. 91.361291.3612
Root an. cond. 3.091653.09165
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 21·5-s − 8·7-s + 8·8-s − 42·10-s + 36·11-s + 49·13-s + 16·14-s − 16·16-s − 42·17-s − 224·19-s − 72·22-s + 180·23-s + 125·25-s − 98·26-s − 135·29-s − 308·31-s + 84·34-s − 168·35-s − 2·37-s + 448·38-s + 168·40-s − 42·41-s − 20·43-s − 360·46-s + 84·47-s + 343·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.87·5-s − 0.431·7-s + 0.353·8-s − 1.32·10-s + 0.986·11-s + 1.04·13-s + 0.305·14-s − 1/4·16-s − 0.599·17-s − 2.70·19-s − 0.697·22-s + 1.63·23-s + 25-s − 0.739·26-s − 0.864·29-s − 1.78·31-s + 0.423·34-s − 0.811·35-s − 0.00888·37-s + 1.91·38-s + 0.664·40-s − 0.159·41-s − 0.0709·43-s − 1.15·46-s + 0.260·47-s + 49-s + ⋯

Functional equation

Λ(s)=(26244s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(26244s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2624426244    =    22⋅382^{2} \cdot 3^{8}
Sign: 11
Analytic conductor: 91.361291.3612
Root analytic conductor: 3.091653.09165
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 26244, ( :3/2,3/2), 1)(4,\ 26244,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.9678211431.967821143
L(12)L(\frac12) ≈\approx 1.9678211431.967821143
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C2C_2 1+pT+p2T2 1 + p T + p^{2} T^{2}
3 1 1
good5C22C_2^2 1−21T+316T2−21p3T3+p6T4 1 - 21 T + 316 T^{2} - 21 p^{3} T^{3} + p^{6} T^{4}
7C22C_2^2 1+8T−279T2+8p3T3+p6T4 1 + 8 T - 279 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4}
11C22C_2^2 1−36T−35T2−36p3T3+p6T4 1 - 36 T - 35 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4}
13C22C_2^2 1−49T+204T2−49p3T3+p6T4 1 - 49 T + 204 T^{2} - 49 p^{3} T^{3} + p^{6} T^{4}
17C2C_2 (1+21T+p3T2)2 ( 1 + 21 T + p^{3} T^{2} )^{2}
19C2C_2 (1+112T+p3T2)2 ( 1 + 112 T + p^{3} T^{2} )^{2}
23C22C_2^2 1−180T+20233T2−180p3T3+p6T4 1 - 180 T + 20233 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4}
29C22C_2^2 1+135T−6164T2+135p3T3+p6T4 1 + 135 T - 6164 T^{2} + 135 p^{3} T^{3} + p^{6} T^{4}
31C2C_2 (1+19T+p3T2)(1+289T+p3T2) ( 1 + 19 T + p^{3} T^{2} )( 1 + 289 T + p^{3} T^{2} )
37C2C_2 (1+T+p3T2)2 ( 1 + T + p^{3} T^{2} )^{2}
41C22C_2^2 1+42T−67157T2+42p3T3+p6T4 1 + 42 T - 67157 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4}
43C22C_2^2 1+20T−79107T2+20p3T3+p6T4 1 + 20 T - 79107 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4}
47C22C_2^2 1−84T−96767T2−84p3T3+p6T4 1 - 84 T - 96767 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4}
53C2C_2 (1−174T+p3T2)2 ( 1 - 174 T + p^{3} T^{2} )^{2}
59C22C_2^2 1−504T+48637T2−504p3T3+p6T4 1 - 504 T + 48637 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4}
61C22C_2^2 1−385T−78756T2−385p3T3+p6T4 1 - 385 T - 78756 T^{2} - 385 p^{3} T^{3} + p^{6} T^{4}
67C22C_2^2 1+272T−226779T2+272p3T3+p6T4 1 + 272 T - 226779 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4}
71C2C_2 (1−888T+p3T2)2 ( 1 - 888 T + p^{3} T^{2} )^{2}
73C2C_2 (1−371T+p3T2)2 ( 1 - 371 T + p^{3} T^{2} )^{2}
79C22C_2^2 1−652T−67935T2−652p3T3+p6T4 1 - 652 T - 67935 T^{2} - 652 p^{3} T^{3} + p^{6} T^{4}
83C22C_2^2 1−84T−564731T2−84p3T3+p6T4 1 - 84 T - 564731 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4}
89C2C_2 (1+21T+p3T2)2 ( 1 + 21 T + p^{3} T^{2} )^{2}
97C22C_2^2 1−1246T+639843T2−1246p3T3+p6T4 1 - 1246 T + 639843 T^{2} - 1246 p^{3} T^{3} + p^{6} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−13.19636071814474684010336587790, −12.37365569412686385313093478142, −11.32444565377788754001933627804, −10.83286639728749434244102639692, −10.79647128807278007024727004397, −9.951693221786258885633588973802, −9.575941390545464730437556384030, −8.972271204568524115182737871001, −8.871044944957653067071524676734, −8.344339804060036266096892806714, −7.27378153000945299767131004090, −6.63646743232648757435686047849, −6.41257781912142378158782433183, −5.74627633160831347732966629430, −5.19645933390127768821127762017, −4.13320893001743525914837271162, −3.61725175184557564453483206474, −2.07105184789143414197317796633, −2.03371887894989969365295511358, −0.75160116229901187165317760534, 0.75160116229901187165317760534, 2.03371887894989969365295511358, 2.07105184789143414197317796633, 3.61725175184557564453483206474, 4.13320893001743525914837271162, 5.19645933390127768821127762017, 5.74627633160831347732966629430, 6.41257781912142378158782433183, 6.63646743232648757435686047849, 7.27378153000945299767131004090, 8.344339804060036266096892806714, 8.871044944957653067071524676734, 8.972271204568524115182737871001, 9.575941390545464730437556384030, 9.951693221786258885633588973802, 10.79647128807278007024727004397, 10.83286639728749434244102639692, 11.32444565377788754001933627804, 12.37365569412686385313093478142, 13.19636071814474684010336587790

Graph of the ZZ-function along the critical line