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Properties

Label 4-3960e2-1.1-c1e2-0-11
Degree 44
Conductor 1568160015681600
Sign 11
Analytic cond. 999.872999.872
Root an. cond. 5.623235.62323
Motivic weight 11
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  − 2·5-s − 4·7-s + 2·11-s + 8·19-s − 8·23-s + 3·25-s + 4·29-s + 4·31-s + 8·35-s − 20·37-s + 4·41-s + 12·43-s − 8·47-s + 4·49-s − 4·53-s − 4·55-s − 4·59-s − 4·61-s + 8·67-s − 12·71-s − 16·73-s − 8·77-s + 16·79-s − 20·83-s − 12·89-s − 16·95-s + 4·97-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s + 0.603·11-s + 1.83·19-s − 1.66·23-s + 3/5·25-s + 0.742·29-s + 0.718·31-s + 1.35·35-s − 3.28·37-s + 0.624·41-s + 1.82·43-s − 1.16·47-s + 4/7·49-s − 0.549·53-s − 0.539·55-s − 0.520·59-s − 0.512·61-s + 0.977·67-s − 1.42·71-s − 1.87·73-s − 0.911·77-s + 1.80·79-s − 2.19·83-s − 1.27·89-s − 1.64·95-s + 0.406·97-s + ⋯

Functional equation

Λ(s)=(15681600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(15681600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1568160015681600    =    26⋅34⋅52⋅1122^{6} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 999.872999.872
Root analytic conductor: 5.623235.62323
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 15681600, ( :1/2,1/2), 1)(4,\ 15681600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
3 1 1
5C1C_1 (1+T)2 ( 1 + T )^{2}
11C1C_1 (1−T)2 ( 1 - T )^{2}
good7D4D_{4} 1+4T+12T2+4pT3+p2T4 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4}
13C22C_2^2 1+20T2+p2T4 1 + 20 T^{2} + p^{2} T^{4}
17C22C_2^2 1−20T2+p2T4 1 - 20 T^{2} + p^{2} T^{4}
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23D4D_{4} 1+8T+38T2+8pT3+p2T4 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4}
29D4D_{4} 1−4T+38T2−4pT3+p2T4 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−4T+42T2−4pT3+p2T4 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4}
37C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
41D4D_{4} 1−4T+62T2−4pT3+p2T4 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−12T+116T2−12pT3+p2T4 1 - 12 T + 116 T^{2} - 12 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+8T+86T2+8pT3+p2T4 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+4T+86T2+4pT3+p2T4 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+4T+98T2+4pT3+p2T4 1 + 4 T + 98 T^{2} + 4 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+4T+102T2+4pT3+p2T4 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4}
67D4D_{4} 1−8T+126T2−8pT3+p2T4 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+12T+154T2+12pT3+p2T4 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+16T+204T2+16pT3+p2T4 1 + 16 T + 204 T^{2} + 16 p T^{3} + p^{2} T^{4}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83D4D_{4} 1+20T+260T2+20pT3+p2T4 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+12T+118T2+12pT3+p2T4 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4}
97D4D_{4} 1−4T−18T2−4pT3+p2T4 1 - 4 T - 18 T^{2} - 4 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

−8.245197891469318402408363034459, −7.943199732467613568871250431352, −7.34635731343169978895671569852, −7.33464826741388030590326826974, −6.70847398775694334947374868371, −6.62219431817092938687080852050, −5.98334254973009386340481409487, −5.87738189451312865979326026818, −5.20833229842077125247464652728, −4.98696135942463515839338435700, −4.21134064005229065019826663617, −4.10265662636460457490231649067, −3.56381535206122722128513294102, −3.26240289590185720383366273231, −2.86935265077217966263767276121, −2.45117204778861715199089250711, −1.43648368552214193129897200113, −1.24351988193469569641720271316, 0, 0, 1.24351988193469569641720271316, 1.43648368552214193129897200113, 2.45117204778861715199089250711, 2.86935265077217966263767276121, 3.26240289590185720383366273231, 3.56381535206122722128513294102, 4.10265662636460457490231649067, 4.21134064005229065019826663617, 4.98696135942463515839338435700, 5.20833229842077125247464652728, 5.87738189451312865979326026818, 5.98334254973009386340481409487, 6.62219431817092938687080852050, 6.70847398775694334947374868371, 7.33464826741388030590326826974, 7.34635731343169978895671569852, 7.943199732467613568871250431352, 8.245197891469318402408363034459

Graph of the ZZ-function along the critical line