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Properties

Label 4-950e2-1.1-c3e2-0-2
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 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 3-s + 12·4-s − 4·6-s + 27·7-s + 32·8-s + 25·9-s − 52·11-s − 12·12-s + 17·13-s + 108·14-s + 80·16-s − 75·17-s + 100·18-s + 38·19-s − 27·21-s − 208·22-s + 203·23-s − 32·24-s + 68·26-s − 76·27-s + 324·28-s + 183·29-s − 18·31-s + 192·32-s + 52·33-s − 300·34-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.192·3-s + 3/2·4-s − 0.272·6-s + 1.45·7-s + 1.41·8-s + 0.925·9-s − 1.42·11-s − 0.288·12-s + 0.362·13-s + 2.06·14-s + 5/4·16-s − 1.07·17-s + 1.30·18-s + 0.458·19-s − 0.280·21-s − 2.01·22-s + 1.84·23-s − 0.272·24-s + 0.512·26-s − 0.541·27-s + 2.18·28-s + 1.17·29-s − 0.104·31-s + 1.06·32-s + 0.274·33-s − 1.51·34-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: 00
Selberg data: (4, 902500, ( :3/2,3/2), 1)(4,\ 902500,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 11.0766759211.07667592
L(12)L(\frac12) ≈\approx 11.0766759211.07667592
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−8pT2+p3T3+p6T4 1 + T - 8 p T^{2} + p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1−27T+790T2−27p3T3+p6T4 1 - 27 T + 790 T^{2} - 27 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+52T+2086T2+52p3T3+p6T4 1 + 52 T + 2086 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−17T+3762T2−17p3T3+p6T4 1 - 17 T + 3762 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+75T+10528T2+75p3T3+p6T4 1 + 75 T + 10528 T^{2} + 75 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−203T+30802T2−203p3T3+p6T4 1 - 203 T + 30802 T^{2} - 203 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−183T+50812T2−183p3T3+p6T4 1 - 183 T + 50812 T^{2} - 183 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+18T+6766T2+18p3T3+p6T4 1 + 18 T + 6766 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+78T+64954T2+78p3T3+p6T4 1 + 78 T + 64954 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4}
41C2C_2 (1−390T+p3T2)2 ( 1 - 390 T + p^{3} T^{2} )^{2}
43D4D_{4} 1+338T+187262T2+338p3T3+p6T4 1 + 338 T + 187262 T^{2} + 338 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+24T+162718T2+24p3T3+p6T4 1 + 24 T + 162718 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−T+77950T2−p3T3+p6T4 1 - T + 77950 T^{2} - p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+687T+370294T2+687p3T3+p6T4 1 + 687 T + 370294 T^{2} + 687 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+1222T+811946T2+1222p3T3+p6T4 1 + 1222 T + 811946 T^{2} + 1222 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−43T+250724T2−43p3T3+p6T4 1 - 43 T + 250724 T^{2} - 43 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−1500T+1176910T2−1500p3T3+p6T4 1 - 1500 T + 1176910 T^{2} - 1500 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−983T+900588T2−983p3T3+p6T4 1 - 983 T + 900588 T^{2} - 983 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−750T+1126390T2−750p3T3+p6T4 1 - 750 T + 1126390 T^{2} - 750 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+1010T+819862T2+1010p3T3+p6T4 1 + 1010 T + 819862 T^{2} + 1010 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−164T+810694T2−164p3T3+p6T4 1 - 164 T + 810694 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−954T+1301362T2−954p3T3+p6T4 1 - 954 T + 1301362 T^{2} - 954 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

−9.970142502673178543521706429951, −9.568975185464477067828508893448, −8.962102782033766914122155121963, −8.515577894766183633104227004063, −7.985851859228781972512507727804, −7.50089586082315143029449144745, −7.48062792106666494278073985435, −6.83372359829170419262197063014, −6.15851405718921442167246758724, −6.11911590948591655652302054274, −5.13464245421887092332178388995, −4.95956734496589551197580896164, −4.77215001675389421853622134617, −4.33082836982738947756603490264, −3.60713698357389011860136956416, −3.05531948206261635909907604664, −2.50328471731616364515862116782, −1.94780221624124080936063720390, −1.35088218372709137864397903339, −0.68115266220540391884769402820, 0.68115266220540391884769402820, 1.35088218372709137864397903339, 1.94780221624124080936063720390, 2.50328471731616364515862116782, 3.05531948206261635909907604664, 3.60713698357389011860136956416, 4.33082836982738947756603490264, 4.77215001675389421853622134617, 4.95956734496589551197580896164, 5.13464245421887092332178388995, 6.11911590948591655652302054274, 6.15851405718921442167246758724, 6.83372359829170419262197063014, 7.48062792106666494278073985435, 7.50089586082315143029449144745, 7.985851859228781972512507727804, 8.515577894766183633104227004063, 8.962102782033766914122155121963, 9.568975185464477067828508893448, 9.970142502673178543521706429951

Graph of the ZZ-function along the critical line