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Properties

Label 4-1520e2-1.1-c1e2-0-26
Degree 44
Conductor 23104002310400
Sign 11
Analytic cond. 147.313147.313
Root an. cond. 3.483853.48385
Motivic weight 11
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  + 4·3-s + 2·5-s + 4·7-s + 8·9-s + 4·11-s − 4·13-s + 8·15-s − 4·17-s + 2·19-s + 16·21-s + 12·23-s + 3·25-s + 12·27-s + 4·29-s + 8·31-s + 16·33-s + 8·35-s − 12·37-s − 16·39-s − 4·41-s − 4·43-s + 16·45-s + 4·47-s + 6·49-s − 16·51-s + 4·53-s + 8·55-s + ⋯
L(s)  = 1  + 2.30·3-s + 0.894·5-s + 1.51·7-s + 8/3·9-s + 1.20·11-s − 1.10·13-s + 2.06·15-s − 0.970·17-s + 0.458·19-s + 3.49·21-s + 2.50·23-s + 3/5·25-s + 2.30·27-s + 0.742·29-s + 1.43·31-s + 2.78·33-s + 1.35·35-s − 1.97·37-s − 2.56·39-s − 0.624·41-s − 0.609·43-s + 2.38·45-s + 0.583·47-s + 6/7·49-s − 2.24·51-s + 0.549·53-s + 1.07·55-s + ⋯

Functional equation

Λ(s)=(2310400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2310400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 23104002310400    =    28⋅52⋅1922^{8} \cdot 5^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 147.313147.313
Root analytic conductor: 3.483853.48385
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2310400, ( :1/2,1/2), 1)(4,\ 2310400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 9.3254699409.325469940
L(12)L(\frac12) ≈\approx 9.3254699409.325469940
L(32)L(\frac{3}{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)
bad2 1 1
5C1C_1 (1−T)2 ( 1 - T )^{2}
19C1C_1 (1−T)2 ( 1 - T )^{2}
good3C22C_2^2 1−4T+8T2−4pT3+p2T4 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4}
7C4C_4 1−4T+10T2−4pT3+p2T4 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4}
11C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
13D4D_{4} 1+4T+12T2+4pT3+p2T4 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4}
17D4D_{4} 1+4T+30T2+4pT3+p2T4 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4}
23C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
29D4D_{4} 1−4T−10T2−4pT3+p2T4 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−8T+70T2−8pT3+p2T4 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+12T+92T2+12pT3+p2T4 1 + 12 T + 92 T^{2} + 12 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+4T+54T2+4pT3+p2T4 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+4T+82T2+4pT3+p2T4 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−4T+90T2−4pT3+p2T4 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−4T+60T2−4pT3+p2T4 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+16T+150T2+16pT3+p2T4 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+16T+154T2+16pT3+p2T4 1 + 16 T + 154 T^{2} + 16 p T^{3} + p^{2} T^{4}
67D4D_{4} 1−4T+136T2−4pT3+p2T4 1 - 4 T + 136 T^{2} - 4 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+16T+198T2+16pT3+p2T4 1 + 16 T + 198 T^{2} + 16 p T^{3} + p^{2} T^{4}
73D4D_{4} 1−12T+110T2−12pT3+p2T4 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−8T+142T2−8pT3+p2T4 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−4T+42T2−4pT3+p2T4 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−4T+110T2−4pT3+p2T4 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+12T+212T2+12pT3+p2T4 1 + 12 T + 212 T^{2} + 12 p T^{3} + p^{2} 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

−9.373346786878501293642312781848, −9.126219677021526773619159829210, −8.889932070403171249287295323269, −8.631701213045925488478188799566, −8.181895377243302681506538839004, −7.83667600992098535447578319865, −7.30155095873499276115561178340, −7.04007419226724166778727163224, −6.55523169887341825854139924299, −6.23951813168676526941690728695, −5.13237721787974012480990535454, −5.07443515478179354900070296507, −4.64366623492648899300369469242, −4.22125217468796483542276910683, −3.35359836181160138157132269303, −3.14706206559333232854953325503, −2.55556377588078431300180226095, −2.22346747470726762631896537131, −1.48431367344935190912877200754, −1.26019537507612455551548251757, 1.26019537507612455551548251757, 1.48431367344935190912877200754, 2.22346747470726762631896537131, 2.55556377588078431300180226095, 3.14706206559333232854953325503, 3.35359836181160138157132269303, 4.22125217468796483542276910683, 4.64366623492648899300369469242, 5.07443515478179354900070296507, 5.13237721787974012480990535454, 6.23951813168676526941690728695, 6.55523169887341825854139924299, 7.04007419226724166778727163224, 7.30155095873499276115561178340, 7.83667600992098535447578319865, 8.181895377243302681506538839004, 8.631701213045925488478188799566, 8.889932070403171249287295323269, 9.126219677021526773619159829210, 9.373346786878501293642312781848

Graph of the ZZ-function along the critical line