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Properties

Label 4-1920e2-1.1-c1e2-0-15
Degree 44
Conductor 36864003686400
Sign 11
Analytic cond. 235.048235.048
Root an. cond. 3.915513.91551
Motivic weight 11
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  − 2·3-s + 2·5-s − 2·7-s + 3·9-s − 4·11-s + 2·13-s − 4·15-s + 6·17-s − 2·19-s + 4·21-s + 2·23-s + 3·25-s − 4·27-s + 6·31-s + 8·33-s − 4·35-s − 2·37-s − 4·39-s − 4·41-s − 4·43-s + 6·45-s + 18·47-s + 6·49-s − 12·51-s + 20·53-s − 8·55-s + 4·57-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 0.755·7-s + 9-s − 1.20·11-s + 0.554·13-s − 1.03·15-s + 1.45·17-s − 0.458·19-s + 0.872·21-s + 0.417·23-s + 3/5·25-s − 0.769·27-s + 1.07·31-s + 1.39·33-s − 0.676·35-s − 0.328·37-s − 0.640·39-s − 0.624·41-s − 0.609·43-s + 0.894·45-s + 2.62·47-s + 6/7·49-s − 1.68·51-s + 2.74·53-s − 1.07·55-s + 0.529·57-s + ⋯

Functional equation

Λ(s)=(3686400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3686400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 36864003686400    =    214⋅32⋅522^{14} \cdot 3^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 235.048235.048
Root analytic conductor: 3.915513.91551
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 3686400, ( :1/2,1/2), 1)(4,\ 3686400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8715668101.871566810
L(12)L(\frac12) ≈\approx 1.8715668101.871566810
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
3C1C_1 (1+T)2 ( 1 + T )^{2}
5C1C_1 (1−T)2 ( 1 - T )^{2}
good7D4D_{4} 1+2T−2T2+2pT3+p2T4 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
11C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
13C4C_4 1−2T+10T2−2pT3+p2T4 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4}
17D4D_{4} 1−6T+26T2−6pT3+p2T4 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4}
19D4D_{4} 1+2T+22T2+2pT3+p2T4 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4}
23D4D_{4} 1−2T+30T2−2pT3+p2T4 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4}
29C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
31D4D_{4} 1−6T+54T2−6pT3+p2T4 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+2T+58T2+2pT3+p2T4 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4}
41C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
43D4D_{4} 1+4T+22T2+4pT3+p2T4 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−18T+158T2−18pT3+p2T4 1 - 18 T + 158 T^{2} - 18 p T^{3} + p^{2} T^{4}
53C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
59C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
61C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
67D4D_{4} 1+4T+70T2+4pT3+p2T4 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4}
71C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
73D4D_{4} 1−8T+94T2−8pT3+p2T4 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−18T+222T2−18pT3+p2T4 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4}
83C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
89C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
97C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
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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

−9.356791594914383908210474172565, −9.122500641723497903450375439740, −8.748323782677769810330679160595, −8.143208932893733349944922815051, −7.78009946351911600510911295873, −7.39536996372988463250358628039, −6.86663253534215083547353363129, −6.60336096654987276107479730905, −6.10346534127001598548807155674, −5.78356093226106515295114625768, −5.45397693751066073370802890689, −5.19060275857337787418167720091, −4.61313099545740155348448603371, −4.15959188512280780902083877018, −3.35398486943687777956529763738, −3.25433193910272785317620247166, −2.31308599493634910666141141733, −2.09642406181533508831481710746, −0.979120241433238298053492313580, −0.68522397140095519720806648681, 0.68522397140095519720806648681, 0.979120241433238298053492313580, 2.09642406181533508831481710746, 2.31308599493634910666141141733, 3.25433193910272785317620247166, 3.35398486943687777956529763738, 4.15959188512280780902083877018, 4.61313099545740155348448603371, 5.19060275857337787418167720091, 5.45397693751066073370802890689, 5.78356093226106515295114625768, 6.10346534127001598548807155674, 6.60336096654987276107479730905, 6.86663253534215083547353363129, 7.39536996372988463250358628039, 7.78009946351911600510911295873, 8.143208932893733349944922815051, 8.748323782677769810330679160595, 9.122500641723497903450375439740, 9.356791594914383908210474172565

Graph of the ZZ-function along the critical line