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Properties

Label 4-1100e2-1.1-c3e2-0-7
Degree 44
Conductor 12100001210000
Sign 11
Analytic cond. 4212.284212.28
Root an. cond. 8.056188.05618
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·3-s − 36·7-s + 18·9-s − 22·11-s + 24·13-s + 8·17-s + 16·19-s − 288·21-s + 312·23-s − 8·27-s − 284·29-s − 432·31-s − 176·33-s − 12·37-s + 192·39-s − 164·41-s − 44·43-s − 152·47-s + 382·49-s + 64·51-s − 124·53-s + 128·57-s − 1.25e3·59-s + 788·61-s − 648·63-s + 752·67-s + 2.49e3·69-s + ⋯
L(s)  = 1  + 1.53·3-s − 1.94·7-s + 2/3·9-s − 0.603·11-s + 0.512·13-s + 0.114·17-s + 0.193·19-s − 2.99·21-s + 2.82·23-s − 0.0570·27-s − 1.81·29-s − 2.50·31-s − 0.928·33-s − 0.0533·37-s + 0.788·39-s − 0.624·41-s − 0.156·43-s − 0.471·47-s + 1.11·49-s + 0.175·51-s − 0.321·53-s + 0.297·57-s − 2.77·59-s + 1.65·61-s − 1.29·63-s + 1.37·67-s + 4.35·69-s + ⋯

Functional equation

Λ(s)=(1210000s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1210000s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 12100001210000    =    24⋅54⋅1122^{4} \cdot 5^{4} \cdot 11^{2}
Sign: 11
Analytic conductor: 4212.284212.28
Root analytic conductor: 8.056188.05618
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 1210000, ( :3/2,3/2), 1)(4,\ 1210000,\ (\ :3/2, 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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
5 1 1
11C1C_1 (1+pT)2 ( 1 + p T )^{2}
good3D4D_{4} 1−8T+46T2−8p3T3+p6T4 1 - 8 T + 46 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1+36T+914T2+36p3T3+p6T4 1 + 36 T + 914 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−24T+3938T2−24p3T3+p6T4 1 - 24 T + 3938 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1−8T−7654T2−8p3T3+p6T4 1 - 8 T - 7654 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1−16T+10326T2−16p3T3+p6T4 1 - 16 T + 10326 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−312T+48646T2−312p3T3+p6T4 1 - 312 T + 48646 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+284T+52718T2+284p3T3+p6T4 1 + 284 T + 52718 T^{2} + 284 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+432T+104702T2+432p3T3+p6T4 1 + 432 T + 104702 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+12T+73598T2+12p3T3+p6T4 1 + 12 T + 73598 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+4pT+130742T2+4p4T3+p6T4 1 + 4 p T + 130742 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4}
43D4D_{4} 1+44T+158634T2+44p3T3+p6T4 1 + 44 T + 158634 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+152T+160406T2+152p3T3+p6T4 1 + 152 T + 160406 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+124T+107198T2+124p3T3+p6T4 1 + 124 T + 107198 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+1256T+745142T2+1256p3T3+p6T4 1 + 1256 T + 745142 T^{2} + 1256 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1−788T+595374T2−788p3T3+p6T4 1 - 788 T + 595374 T^{2} - 788 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−752T+730206T2−752p3T3+p6T4 1 - 752 T + 730206 T^{2} - 752 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+1312T+1035182T2+1312p3T3+p6T4 1 + 1312 T + 1035182 T^{2} + 1312 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+1480T+1322730T2+1480p3T3+p6T4 1 + 1480 T + 1322730 T^{2} + 1480 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+40T+488814T2+40p3T3+p6T4 1 + 40 T + 488814 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+1068T+1428634T2+1068p3T3+p6T4 1 + 1068 T + 1428634 T^{2} + 1068 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+132T−745706T2+132p3T3+p6T4 1 + 132 T - 745706 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1+444T+1874534T2+444p3T3+p6T4 1 + 444 T + 1874534 T^{2} + 444 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

−9.236852287269372570298592556047, −8.868255018603558898881681780505, −8.697990505102643742249891072183, −8.128367944863051323297909071199, −7.50438161268950006861414663283, −7.28029036160500912654170237694, −6.91437767365896136874890744025, −6.47058956638088605281649281077, −5.76813618761712496432761235672, −5.53756718138606076680279323398, −4.99495174708360247571941581753, −4.25179585954205962223125668975, −3.53343015992190064628514531595, −3.40691031488742424185551637381, −2.91430133487395756053670041068, −2.76350554130163053658792587797, −1.82086466567617497167348347900, −1.29005186019088573770596383251, 0, 0, 1.29005186019088573770596383251, 1.82086466567617497167348347900, 2.76350554130163053658792587797, 2.91430133487395756053670041068, 3.40691031488742424185551637381, 3.53343015992190064628514531595, 4.25179585954205962223125668975, 4.99495174708360247571941581753, 5.53756718138606076680279323398, 5.76813618761712496432761235672, 6.47058956638088605281649281077, 6.91437767365896136874890744025, 7.28029036160500912654170237694, 7.50438161268950006861414663283, 8.128367944863051323297909071199, 8.697990505102643742249891072183, 8.868255018603558898881681780505, 9.236852287269372570298592556047

Graph of the ZZ-function along the critical line