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Properties

Label 4-950e2-1.1-c3e2-0-8
Degree 44
Conductor 902500902500
Sign 11
Analytic cond. 3141.803141.80
Root an. cond. 7.486777.48677
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  + 4·2-s − 2·3-s + 12·4-s − 8·6-s − 8·7-s + 32·8-s − 24·9-s + 28·11-s − 24·12-s − 2·13-s − 32·14-s + 80·16-s − 152·17-s − 96·18-s − 38·19-s + 16·21-s + 112·22-s − 284·23-s − 64·24-s − 8·26-s + 50·27-s − 96·28-s + 144·29-s + 128·31-s + 192·32-s − 56·33-s − 608·34-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.384·3-s + 3/2·4-s − 0.544·6-s − 0.431·7-s + 1.41·8-s − 8/9·9-s + 0.767·11-s − 0.577·12-s − 0.0426·13-s − 0.610·14-s + 5/4·16-s − 2.16·17-s − 1.25·18-s − 0.458·19-s + 0.166·21-s + 1.08·22-s − 2.57·23-s − 0.544·24-s − 0.0603·26-s + 0.356·27-s − 0.647·28-s + 0.922·29-s + 0.741·31-s + 1.06·32-s − 0.295·33-s − 3.06·34-s + ⋯

Functional equation

Λ(s)=(902500s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(902500s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 902500902500    =    22⋅54⋅1922^{2} \cdot 5^{4} \cdot 19^{2}
Sign: 11
Analytic conductor: 3141.803141.80
Root analytic conductor: 7.486777.48677
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 902500, ( :3/2,3/2), 1)(4,\ 902500,\ (\ :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)
bad2C1C_1 (1−pT)2 ( 1 - p T )^{2}
5 1 1
19C1C_1 (1+pT)2 ( 1 + p T )^{2}
good3D4D_{4} 1+2T+28T2+2p3T3+p6T4 1 + 2 T + 28 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1+8T+510T2+8p3T3+p6T4 1 + 8 T + 510 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1−28T+1658T2−28p3T3+p6T4 1 - 28 T + 1658 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1+2T−648T2+2p3T3+p6T4 1 + 2 T - 648 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+152T+14630T2+152p3T3+p6T4 1 + 152 T + 14630 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+284T+43910T2+284p3T3+p6T4 1 + 284 T + 43910 T^{2} + 284 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−144T+49630T2−144p3T3+p6T4 1 - 144 T + 49630 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−128T+38286T2−128p3T3+p6T4 1 - 128 T + 38286 T^{2} - 128 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−194T+104640T2−194p3T3+p6T4 1 - 194 T + 104640 T^{2} - 194 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−608T+213830T2−608p3T3+p6T4 1 - 608 T + 213830 T^{2} - 608 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+288T+175862T2+288p3T3+p6T4 1 + 288 T + 175862 T^{2} + 288 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+432T+188590T2+432p3T3+p6T4 1 + 432 T + 188590 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+810T+453352T2+810p3T3+p6T4 1 + 810 T + 453352 T^{2} + 810 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−324T+333214T2−324p3T3+p6T4 1 - 324 T + 333214 T^{2} - 324 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+1076T+698754T2+1076p3T3+p6T4 1 + 1076 T + 698754 T^{2} + 1076 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+658T+661380T2+658p3T3+p6T4 1 + 658 T + 661380 T^{2} + 658 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+1696T+1433198T2+1696p3T3+p6T4 1 + 1696 T + 1433198 T^{2} + 1696 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+832T+738822T2+832p3T3+p6T4 1 + 832 T + 738822 T^{2} + 832 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+1428T+1222262T2+1428p3T3+p6T4 1 + 1428 T + 1222262 T^{2} + 1428 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−508T+848942T2−508p3T3+p6T4 1 - 508 T + 848942 T^{2} - 508 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−576T+448582T2−576p3T3+p6T4 1 - 576 T + 448582 T^{2} - 576 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1+834T+1999160T2+834p3T3+p6T4 1 + 834 T + 1999160 T^{2} + 834 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

−9.448959636667590691187629706141, −9.087615638466382584048561414803, −8.451647224305470413009014459005, −8.260456193940835627131396145190, −7.59837756726843962762047557598, −7.21978367490010068882005092533, −6.41631033986201676233939676517, −6.28447917937203411615459605031, −6.08687681922474799801208273142, −5.80240638924554050967754806514, −4.67312451042721037812684850559, −4.59794372601258761984655648470, −4.33650191591405446968441595898, −3.65687137884615326326368383470, −2.91833349582925394225778191131, −2.72858653096269150697097643602, −1.91767739377656276507970096672, −1.45978496085265755233525692299, 0, 0, 1.45978496085265755233525692299, 1.91767739377656276507970096672, 2.72858653096269150697097643602, 2.91833349582925394225778191131, 3.65687137884615326326368383470, 4.33650191591405446968441595898, 4.59794372601258761984655648470, 4.67312451042721037812684850559, 5.80240638924554050967754806514, 6.08687681922474799801208273142, 6.28447917937203411615459605031, 6.41631033986201676233939676517, 7.21978367490010068882005092533, 7.59837756726843962762047557598, 8.260456193940835627131396145190, 8.451647224305470413009014459005, 9.087615638466382584048561414803, 9.448959636667590691187629706141

Graph of the ZZ-function along the critical line