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Properties

Label 4-120e2-1.1-c1e2-0-1
Degree 44
Conductor 1440014400
Sign 11
Analytic cond. 0.9181560.918156
Root an. cond. 0.9788790.978879
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-s − 2·3-s − 4-s + 2·6-s + 3·8-s + 3·9-s − 8·11-s + 2·12-s − 16-s + 4·17-s − 3·18-s + 8·19-s + 8·22-s − 6·24-s + 25-s − 4·27-s − 5·32-s + 16·33-s − 4·34-s − 3·36-s − 8·38-s + 20·41-s + 8·43-s + 8·44-s + 2·48-s − 14·49-s − 50-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s + 1.06·8-s + 9-s − 2.41·11-s + 0.577·12-s − 1/4·16-s + 0.970·17-s − 0.707·18-s + 1.83·19-s + 1.70·22-s − 1.22·24-s + 1/5·25-s − 0.769·27-s − 0.883·32-s + 2.78·33-s − 0.685·34-s − 1/2·36-s − 1.29·38-s + 3.12·41-s + 1.21·43-s + 1.20·44-s + 0.288·48-s − 2·49-s − 0.141·50-s + ⋯

Functional equation

Λ(s)=(14400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(14400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1440014400    =    26⋅32⋅522^{6} \cdot 3^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 0.9181560.918156
Root analytic conductor: 0.9788790.978879
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 14400, ( :1/2,1/2), 1)(4,\ 14400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.39521996040.3952199604
L(12)L(\frac12) ≈\approx 0.39521996040.3952199604
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)
bad2C2C_2 1+T+pT2 1 + T + p T^{2}
3C1C_1 (1+T)2 ( 1 + T )^{2}
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
41C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
43C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
47C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
59C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
61C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
67C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
71C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
73C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C2C_2 (1−2T+pT2)2 ( 1 - 2 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

−10.98983893146666191058257315512, −10.67892245123374144028858824682, −10.09004766089519189801221596346, −9.523451675812284265792380668213, −9.339349995423015885465559763611, −8.074746559817531846840486780032, −7.67749235610857150327119620769, −7.66488013441745380243230842523, −6.56365355473852823699216167747, −5.63406482120703248001185629214, −5.23920392624592057055772361749, −4.87700802399738744801917392626, −3.76558570458913747879946495025, −2.61544872003966305633910980827, −0.870154925974925884967934945032, 0.870154925974925884967934945032, 2.61544872003966305633910980827, 3.76558570458913747879946495025, 4.87700802399738744801917392626, 5.23920392624592057055772361749, 5.63406482120703248001185629214, 6.56365355473852823699216167747, 7.66488013441745380243230842523, 7.67749235610857150327119620769, 8.074746559817531846840486780032, 9.339349995423015885465559763611, 9.523451675812284265792380668213, 10.09004766089519189801221596346, 10.67892245123374144028858824682, 10.98983893146666191058257315512

Graph of the ZZ-function along the critical line