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Properties

Label 4-525e2-1.1-c3e2-0-13
Degree 44
Conductor 275625275625
Sign 11
Analytic cond. 959.512959.512
Root an. cond. 5.565605.56560
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  − 3·2-s + 6·3-s − 5·4-s − 18·6-s + 14·7-s + 33·8-s + 27·9-s − 31·11-s − 30·12-s − 39·13-s − 42·14-s − 21·16-s + 79·17-s − 81·18-s − 56·19-s + 84·21-s + 93·22-s − 254·23-s + 198·24-s + 117·26-s + 108·27-s − 70·28-s − 62·29-s − 135·31-s − 87·32-s − 186·33-s − 237·34-s + ⋯
L(s)  = 1  − 1.06·2-s + 1.15·3-s − 5/8·4-s − 1.22·6-s + 0.755·7-s + 1.45·8-s + 9-s − 0.849·11-s − 0.721·12-s − 0.832·13-s − 0.801·14-s − 0.328·16-s + 1.12·17-s − 1.06·18-s − 0.676·19-s + 0.872·21-s + 0.901·22-s − 2.30·23-s + 1.68·24-s + 0.882·26-s + 0.769·27-s − 0.472·28-s − 0.397·29-s − 0.782·31-s − 0.480·32-s − 0.981·33-s − 1.19·34-s + ⋯

Functional equation

Λ(s)=(275625s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(275625s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 275625275625    =    32⋅54⋅723^{2} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 959.512959.512
Root analytic conductor: 5.565605.56560
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 275625, ( :3/2,3/2), 1)(4,\ 275625,\ (\ :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)
bad3C1C_1 (1−pT)2 ( 1 - p T )^{2}
5 1 1
7C1C_1 (1−pT)2 ( 1 - p T )^{2}
good2C4C_4 1+3T+7pT2+3p3T3+p6T4 1 + 3 T + 7 p T^{2} + 3 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+31T+2796T2+31p3T3+p6T4 1 + 31 T + 2796 T^{2} + 31 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1+3pT+3546T2+3p4T3+p6T4 1 + 3 p T + 3546 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4}
17D4D_{4} 1−79T+8288T2−79p3T3+p6T4 1 - 79 T + 8288 T^{2} - 79 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+56T+1174T2+56p3T3+p6T4 1 + 56 T + 1174 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+254T+38015T2+254p3T3+p6T4 1 + 254 T + 38015 T^{2} + 254 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+62T+19751T2+62p3T3+p6T4 1 + 62 T + 19751 T^{2} + 62 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+135T+63420T2+135p3T3+p6T4 1 + 135 T + 63420 T^{2} + 135 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+113T+102964T2+113p3T3+p6T4 1 + 113 T + 102964 T^{2} + 113 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−235T+143042T2−235p3T3+p6T4 1 - 235 T + 143042 T^{2} - 235 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+804T+306321T2+804p3T3+p6T4 1 + 804 T + 306321 T^{2} + 804 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+152T+180510T2+152p3T3+p6T4 1 + 152 T + 180510 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+149T+269640T2+149p3T3+p6T4 1 + 149 T + 269640 T^{2} + 149 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+441T+273734T2+441p3T3+p6T4 1 + 441 T + 273734 T^{2} + 441 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+223T+149646T2+223p3T3+p6T4 1 + 223 T + 149646 T^{2} + 223 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+1157T+871890T2+1157p3T3+p6T4 1 + 1157 T + 871890 T^{2} + 1157 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−619T+576906T2−619p3T3+p6T4 1 - 619 T + 576906 T^{2} - 619 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+268T+763078T2+268p3T3+p6T4 1 + 268 T + 763078 T^{2} + 268 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+427T+986572T2+427p3T3+p6T4 1 + 427 T + 986572 T^{2} + 427 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+1211T+720720T2+1211p3T3+p6T4 1 + 1211 T + 720720 T^{2} + 1211 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−466T+306170T2−466p3T3+p6T4 1 - 466 T + 306170 T^{2} - 466 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1+172T−273626T2+172p3T3+p6T4 1 + 172 T - 273626 T^{2} + 172 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

−9.978844281186729085710753821351, −9.849708628900840625392888038927, −9.182752734382268522602132133826, −9.015677661701696983659479578508, −8.241889097431895984297069870114, −8.189074437725777677509591503270, −7.69531193756662675881230856221, −7.63069371584256306352029705758, −6.80038337320182259248395398509, −6.12807000959301484725777218404, −5.24431720393011742867523080898, −5.16679298048263652568632294577, −4.17748629582954838913062794466, −4.16619229371580336296790025746, −3.23530014965051729462661896334, −2.61924209101489875657644639319, −1.73211390683513546126124327273, −1.50634273801803250279664283532, 0, 0, 1.50634273801803250279664283532, 1.73211390683513546126124327273, 2.61924209101489875657644639319, 3.23530014965051729462661896334, 4.16619229371580336296790025746, 4.17748629582954838913062794466, 5.16679298048263652568632294577, 5.24431720393011742867523080898, 6.12807000959301484725777218404, 6.80038337320182259248395398509, 7.63069371584256306352029705758, 7.69531193756662675881230856221, 8.189074437725777677509591503270, 8.241889097431895984297069870114, 9.015677661701696983659479578508, 9.182752734382268522602132133826, 9.849708628900840625392888038927, 9.978844281186729085710753821351

Graph of the ZZ-function along the critical line