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Properties

Label 4-3240e2-1.1-c0e2-0-9
Degree 44
Conductor 1049760010497600
Sign 11
Analytic cond. 2.614592.61459
Root an. cond. 1.271601.27160
Motivic weight 00
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 + 5-s + 8-s − 10-s − 16-s + 2·19-s − 23-s + 3·31-s − 2·38-s + 40-s + 46-s − 2·47-s + 49-s − 2·53-s − 3·61-s − 3·62-s + 64-s + 3·79-s − 80-s + 3·83-s + 2·94-s + 2·95-s − 98-s + 2·106-s − 115-s + 121-s + 3·122-s + ⋯
L(s)  = 1  − 2-s + 5-s + 8-s − 10-s − 16-s + 2·19-s − 23-s + 3·31-s − 2·38-s + 40-s + 46-s − 2·47-s + 49-s − 2·53-s − 3·61-s − 3·62-s + 64-s + 3·79-s − 80-s + 3·83-s + 2·94-s + 2·95-s − 98-s + 2·106-s − 115-s + 121-s + 3·122-s + ⋯

Functional equation

Λ(s)=(10497600s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(10497600s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1049760010497600    =    26⋅38⋅522^{6} \cdot 3^{8} \cdot 5^{2}
Sign: 11
Analytic conductor: 2.614592.61459
Root analytic conductor: 1.271601.27160
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 10497600, ( :0,0), 1)(4,\ 10497600,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.98471616320.9847161632
L(12)L(\frac12) ≈\approx 0.98471616320.9847161632
L(1)L(1) 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+T2 1 + T + T^{2}
3 1 1
5C2C_2 1−T+T2 1 - T + T^{2}
good7C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
11C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
13C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
17C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
19C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
23C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
29C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
31C1C_1×\timesC2C_2 (1−T)2(1−T+T2) ( 1 - T )^{2}( 1 - T + T^{2} )
37C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
41C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
43C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
47C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
53C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
59C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
61C1C_1×\timesC2C_2 (1+T)2(1+T+T2) ( 1 + T )^{2}( 1 + T + T^{2} )
67C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
71C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
73C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
79C1C_1×\timesC2C_2 (1−T)2(1−T+T2) ( 1 - T )^{2}( 1 - T + T^{2} )
83C1C_1×\timesC2C_2 (1−T)2(1−T+T2) ( 1 - T )^{2}( 1 - T + T^{2} )
89C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
97C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{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

−9.114380424932356413029279956265, −8.774639717621336695108187650658, −8.063613244514016110045635227022, −7.964293178622560157125849434103, −7.83330835762579070760500838464, −7.35701975827551266653767395344, −6.62366266728923752217915642105, −6.52888064506062083826810214654, −6.09157546690649956605185555807, −5.74538172411086929913050377280, −5.07868091854618054292975873012, −4.84980638737296705692982102553, −4.60308608784609179346475750161, −3.95428718467989718258389281024, −3.24176508697820678014290675531, −3.12133694147074461780703827854, −2.33271783431510951357366550408, −1.87788608412848438253145007340, −1.35224497254611265376290807971, −0.78878678362419557792982122383, 0.78878678362419557792982122383, 1.35224497254611265376290807971, 1.87788608412848438253145007340, 2.33271783431510951357366550408, 3.12133694147074461780703827854, 3.24176508697820678014290675531, 3.95428718467989718258389281024, 4.60308608784609179346475750161, 4.84980638737296705692982102553, 5.07868091854618054292975873012, 5.74538172411086929913050377280, 6.09157546690649956605185555807, 6.52888064506062083826810214654, 6.62366266728923752217915642105, 7.35701975827551266653767395344, 7.83330835762579070760500838464, 7.964293178622560157125849434103, 8.063613244514016110045635227022, 8.774639717621336695108187650658, 9.114380424932356413029279956265

Graph of the ZZ-function along the critical line