Properties

Label 4-650e2-1.1-c3e2-0-9
Degree 44
Conductor 422500422500
Sign 11
Analytic cond. 1470.811470.81
Root an. cond. 6.192836.19283
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  − 4·2-s + 7·3-s + 12·4-s − 28·6-s + 5·7-s − 32·8-s + 9·9-s − 51·11-s + 84·12-s − 26·13-s − 20·14-s + 80·16-s + 63·17-s − 36·18-s + 28·19-s + 35·21-s + 204·22-s + 9·23-s − 224·24-s + 104·26-s − 28·27-s + 60·28-s − 198·29-s + 175·31-s − 192·32-s − 357·33-s − 252·34-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.34·3-s + 3/2·4-s − 1.90·6-s + 0.269·7-s − 1.41·8-s + 1/3·9-s − 1.39·11-s + 2.02·12-s − 0.554·13-s − 0.381·14-s + 5/4·16-s + 0.898·17-s − 0.471·18-s + 0.338·19-s + 0.363·21-s + 1.97·22-s + 0.0815·23-s − 1.90·24-s + 0.784·26-s − 0.199·27-s + 0.404·28-s − 1.26·29-s + 1.01·31-s − 1.06·32-s − 1.88·33-s − 1.27·34-s + ⋯

Functional equation

Λ(s)=(422500s/2ΓC(s)2L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(422500s/2ΓC(s+3/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 422500422500    =    22541322^{2} \cdot 5^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 1470.811470.81
Root analytic conductor: 6.192836.19283
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 422500, ( :3/2,3/2), 1)(4,\ 422500,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) \approx 1.9095354021.909535402
L(12)L(\frac12) \approx 1.9095354021.909535402
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)
bad2C1C_1 (1+pT)2 ( 1 + p T )^{2}
5 1 1
13C1C_1 (1+pT)2 ( 1 + p T )^{2}
good3D4D_{4} 17T+40T27p3T3+p6T4 1 - 7 T + 40 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 15T+666T25p3T3+p6T4 1 - 5 T + 666 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+51T+2656T2+51p3T3+p6T4 1 + 51 T + 2656 T^{2} + 51 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 163T+9532T263p3T3+p6T4 1 - 63 T + 9532 T^{2} - 63 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 128T+10134T228p3T3+p6T4 1 - 28 T + 10134 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 19T+16768T29p3T3+p6T4 1 - 9 T + 16768 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+198T+56899T2+198p3T3+p6T4 1 + 198 T + 56899 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1175T+64062T2175p3T3+p6T4 1 - 175 T + 64062 T^{2} - 175 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1632T+197382T2632p3T3+p6T4 1 - 632 T + 197382 T^{2} - 632 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1210T+136162T2210p3T3+p6T4 1 - 210 T + 136162 T^{2} - 210 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 123T+151560T223p3T3+p6T4 1 - 23 T + 151560 T^{2} - 23 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1231T+129610T2231p3T3+p6T4 1 - 231 T + 129610 T^{2} - 231 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+186T+285823T2+186p3T3+p6T4 1 + 186 T + 285823 T^{2} + 186 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 163T+410464T263p3T3+p6T4 1 - 63 T + 410464 T^{2} - 63 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+182T+240063T2+182p3T3+p6T4 1 + 182 T + 240063 T^{2} + 182 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1695T+523596T2695p3T3+p6T4 1 - 695 T + 523596 T^{2} - 695 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1390T193778T2390p3T3+p6T4 1 - 390 T - 193778 T^{2} - 390 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 11526T+1231578T21526p3T3+p6T4 1 - 1526 T + 1231578 T^{2} - 1526 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+863T+991434T2+863p3T3+p6T4 1 + 863 T + 991434 T^{2} + 863 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1315T+585604T2315p3T3+p6T4 1 - 315 T + 585604 T^{2} - 315 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+1638T+1580794T2+1638p3T3+p6T4 1 + 1638 T + 1580794 T^{2} + 1638 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 198T+996042T298p3T3+p6T4 1 - 98 T + 996042 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4}
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   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

−10.14457539118605069147347330741, −9.668300201491275622154802080968, −9.426879682177825617975149046077, −9.323951962850405555687678826758, −8.344792590739361304099178245460, −8.249028020448431955275621115210, −7.88291219545671808973735537601, −7.72687333271058219638906277977, −7.11070658147207233406910946179, −6.63577798542684304804278507107, −5.72149247477053852037659481783, −5.71167017401484748595428607984, −4.82751247384841894313216212725, −4.27671325650200537056196266278, −3.17863618884491847534005883639, −3.17215838905741677826473202020, −2.29552048444035712697478301742, −2.23989294362442183115129447094, −1.12651675474771976048296918426, −0.49776152233045255375828449596, 0.49776152233045255375828449596, 1.12651675474771976048296918426, 2.23989294362442183115129447094, 2.29552048444035712697478301742, 3.17215838905741677826473202020, 3.17863618884491847534005883639, 4.27671325650200537056196266278, 4.82751247384841894313216212725, 5.71167017401484748595428607984, 5.72149247477053852037659481783, 6.63577798542684304804278507107, 7.11070658147207233406910946179, 7.72687333271058219638906277977, 7.88291219545671808973735537601, 8.249028020448431955275621115210, 8.344792590739361304099178245460, 9.323951962850405555687678826758, 9.426879682177825617975149046077, 9.668300201491275622154802080968, 10.14457539118605069147347330741

Graph of the ZZ-function along the critical line