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Properties

Label 4-950e2-1.1-c3e2-0-7
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 − 3-s + 12·4-s − 4·6-s − 57·7-s + 32·8-s − 9·9-s + 10·11-s − 12·12-s − 13·13-s − 228·14-s + 80·16-s + 51·17-s − 36·18-s − 38·19-s + 57·21-s + 40·22-s + 155·23-s − 32·24-s − 52·26-s − 8·27-s − 684·28-s − 79·29-s − 16·31-s + 192·32-s − 10·33-s + 204·34-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.192·3-s + 3/2·4-s − 0.272·6-s − 3.07·7-s + 1.41·8-s − 1/3·9-s + 0.274·11-s − 0.288·12-s − 0.277·13-s − 4.35·14-s + 5/4·16-s + 0.727·17-s − 0.471·18-s − 0.458·19-s + 0.592·21-s + 0.387·22-s + 1.40·23-s − 0.272·24-s − 0.392·26-s − 0.0570·27-s − 4.61·28-s − 0.505·29-s − 0.0926·31-s + 1.06·32-s − 0.0527·33-s + 1.02·34-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+T+10T2+p3T3+p6T4 1 + T + 10 T^{2} + p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1+57T+1454T2+57p3T3+p6T4 1 + 57 T + 1454 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1−10T+2510T2−10p3T3+p6T4 1 - 10 T + 2510 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1+pT+2268T2+p4T3+p6T4 1 + p T + 2268 T^{2} + p^{4} T^{3} + p^{6} T^{4}
17D4D_{4} 1−3pT+520T2−3p4T3+p6T4 1 - 3 p T + 520 T^{2} - 3 p^{4} T^{3} + p^{6} T^{4}
23D4D_{4} 1−155T+994pT2−155p3T3+p6T4 1 - 155 T + 994 p T^{2} - 155 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+79T+13124T2+79p3T3+p6T4 1 + 79 T + 13124 T^{2} + 79 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+16T+48318T2+16p3T3+p6T4 1 + 16 T + 48318 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+380T+126078T2+380p3T3+p6T4 1 + 380 T + 126078 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+790T+292274T2+790p3T3+p6T4 1 + 790 T + 292274 T^{2} + 790 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+296T+78966T2+296p3T3+p6T4 1 + 296 T + 78966 T^{2} + 296 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1−200T+146846T2−200p3T3+p6T4 1 - 200 T + 146846 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+397T+333572T2+397p3T3+p6T4 1 + 397 T + 333572 T^{2} + 397 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−201T+197794T2−201p3T3+p6T4 1 - 201 T + 197794 T^{2} - 201 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+680T+483894T2+680p3T3+p6T4 1 + 680 T + 483894 T^{2} + 680 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−939T+740138T2−939p3T3+p6T4 1 - 939 T + 740138 T^{2} - 939 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−406T+735614T2−406p3T3+p6T4 1 - 406 T + 735614 T^{2} - 406 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+123T+781772T2+123p3T3+p6T4 1 + 123 T + 781772 T^{2} + 123 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−106T+840030T2−106p3T3+p6T4 1 - 106 T + 840030 T^{2} - 106 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+2226T+2380750T2+2226p3T3+p6T4 1 + 2226 T + 2380750 T^{2} + 2226 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+870T+1594738T2+870p3T3+p6T4 1 + 870 T + 1594738 T^{2} + 870 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−1864T+2438382T2−1864p3T3+p6T4 1 - 1864 T + 2438382 T^{2} - 1864 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

−9.629576398992878818528042059212, −9.194662380355910187118361404389, −8.525297720716282184656598593212, −8.342474162730769681284093307756, −7.30597435772406113342176953640, −6.93934498463053888675641504050, −6.88083544045976845019605304651, −6.46577391308989674088085143348, −5.84585713140762792056314855940, −5.69854559396613702682872783528, −5.07269518462082386258986104834, −4.66408454669520975301503332128, −3.72840049233367941783150918293, −3.56001818230604933785408208542, −3.14949896343821601494171024124, −2.87917478794303451032666866285, −2.05745957732913278040109158479, −1.24035554946339237114310749178, 0, 0, 1.24035554946339237114310749178, 2.05745957732913278040109158479, 2.87917478794303451032666866285, 3.14949896343821601494171024124, 3.56001818230604933785408208542, 3.72840049233367941783150918293, 4.66408454669520975301503332128, 5.07269518462082386258986104834, 5.69854559396613702682872783528, 5.84585713140762792056314855940, 6.46577391308989674088085143348, 6.88083544045976845019605304651, 6.93934498463053888675641504050, 7.30597435772406113342176953640, 8.342474162730769681284093307756, 8.525297720716282184656598593212, 9.194662380355910187118361404389, 9.629576398992878818528042059212

Graph of the ZZ-function along the critical line