Properties

Label 4-704e2-1.1-c3e2-0-8
Degree 44
Conductor 495616495616
Sign 11
Analytic cond. 1725.351725.35
Root an. cond. 6.444946.44494
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·5-s + 20·7-s − 3·9-s + 22·11-s − 80·13-s − 4·15-s − 124·17-s − 72·19-s + 40·21-s − 98·23-s − 55·25-s + 34·27-s − 144·29-s − 34·31-s + 44·33-s − 40·35-s − 54·37-s − 160·39-s + 536·41-s + 60·43-s + 6·45-s − 272·47-s − 338·49-s − 248·51-s + 492·53-s − 44·55-s + ⋯
L(s)  = 1  + 0.384·3-s − 0.178·5-s + 1.07·7-s − 1/9·9-s + 0.603·11-s − 1.70·13-s − 0.0688·15-s − 1.76·17-s − 0.869·19-s + 0.415·21-s − 0.888·23-s − 0.439·25-s + 0.242·27-s − 0.922·29-s − 0.196·31-s + 0.232·33-s − 0.193·35-s − 0.239·37-s − 0.656·39-s + 2.04·41-s + 0.212·43-s + 0.0198·45-s − 0.844·47-s − 0.985·49-s − 0.680·51-s + 1.27·53-s − 0.107·55-s + ⋯

Functional equation

Λ(s)=(495616s/2ΓC(s)2L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(495616s/2ΓC(s+3/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 495616495616    =    2121122^{12} \cdot 11^{2}
Sign: 11
Analytic conductor: 1725.351725.35
Root analytic conductor: 6.444946.44494
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 495616, ( :3/2,3/2), 1)(4,\ 495616,\ (\ :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(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
11C1C_1 (1pT)2 ( 1 - p T )^{2}
good3D4D_{4} 12T+7T22p3T3+p6T4 1 - 2 T + 7 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4}
5D4D_{4} 1+2T+59T2+2p3T3+p6T4 1 + 2 T + 59 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 120T+738T220p3T3+p6T4 1 - 20 T + 738 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1+80T+4794T2+80p3T3+p6T4 1 + 80 T + 4794 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+124T+13238T2+124p3T3+p6T4 1 + 124 T + 13238 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+72T+4214T2+72p3T3+p6T4 1 + 72 T + 4214 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+98T+22847T2+98p3T3+p6T4 1 + 98 T + 22847 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+144T+44554T2+144p3T3+p6T4 1 + 144 T + 44554 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+34T+57519T2+34p3T3+p6T4 1 + 34 T + 57519 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+54T+101843T2+54p3T3+p6T4 1 + 54 T + 101843 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1536T+209618T2536p3T3+p6T4 1 - 536 T + 209618 T^{2} - 536 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 160T+159146T260p3T3+p6T4 1 - 60 T + 159146 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+272T+182942T2+272p3T3+p6T4 1 + 272 T + 182942 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1492T+348862T2492p3T3+p6T4 1 - 492 T + 348862 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+634T+458975T2+634p3T3+p6T4 1 + 634 T + 458975 T^{2} + 634 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+840T+528794T2+840p3T3+p6T4 1 + 840 T + 528794 T^{2} + 840 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+754T+742455T2+754p3T3+p6T4 1 + 754 T + 742455 T^{2} + 754 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+678T+813415T2+678p3T3+p6T4 1 + 678 T + 813415 T^{2} + 678 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+400T+160962T2+400p3T3+p6T4 1 + 400 T + 160962 T^{2} + 400 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 14pT279966T24p4T3+p6T4 1 - 4 p T - 279966 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4}
83D4D_{4} 1+468T+1155130T2+468p3T3+p6T4 1 + 468 T + 1155130 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+1842T+1935427T2+1842p3T3+p6T4 1 + 1842 T + 1935427 T^{2} + 1842 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 12194T+2966547T22194p3T3+p6T4 1 - 2194 T + 2966547 T^{2} - 2194 p^{3} T^{3} + p^{6} T^{4}
show more
show less
   L(s)=p j=14(1αj,pps)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

−9.659890211653806100668861196911, −9.337001964357384850338755548042, −9.021088455503012753060679925998, −8.549173266722828068742926250328, −8.068786073483896527883817655673, −7.67320478042031699270768780324, −7.33516177957734884300535111967, −6.82673914574715326162010760868, −6.28038038251687987262360181879, −5.82693718417527506000725229542, −5.22613680538853857620486748250, −4.45217106719644271208490575739, −4.45122371502964288719910280408, −4.00134385091175629378275306062, −2.94621839644524322570989992619, −2.55201582993080766996807852719, −1.86280330592618412142484737768, −1.52242411151265508876516760398, 0, 0, 1.52242411151265508876516760398, 1.86280330592618412142484737768, 2.55201582993080766996807852719, 2.94621839644524322570989992619, 4.00134385091175629378275306062, 4.45122371502964288719910280408, 4.45217106719644271208490575739, 5.22613680538853857620486748250, 5.82693718417527506000725229542, 6.28038038251687987262360181879, 6.82673914574715326162010760868, 7.33516177957734884300535111967, 7.67320478042031699270768780324, 8.068786073483896527883817655673, 8.549173266722828068742926250328, 9.021088455503012753060679925998, 9.337001964357384850338755548042, 9.659890211653806100668861196911

Graph of the ZZ-function along the critical line