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Properties

Label 4-3240e2-1.1-c0e2-0-13
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  − 2·2-s + 3·4-s + 2·5-s − 4·8-s − 4·10-s + 5·16-s + 2·19-s + 6·20-s − 2·23-s + 3·25-s − 6·32-s − 4·38-s − 8·40-s + 4·46-s + 2·47-s + 49-s − 6·50-s + 2·53-s + 7·64-s + 6·76-s + 10·80-s − 6·92-s − 4·94-s + 4·95-s − 2·98-s + 9·100-s − 4·106-s + ⋯
L(s)  = 1  − 2·2-s + 3·4-s + 2·5-s − 4·8-s − 4·10-s + 5·16-s + 2·19-s + 6·20-s − 2·23-s + 3·25-s − 6·32-s − 4·38-s − 8·40-s + 4·46-s + 2·47-s + 49-s − 6·50-s + 2·53-s + 7·64-s + 6·76-s + 10·80-s − 6·92-s − 4·94-s + 4·95-s − 2·98-s + 9·100-s − 4·106-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 1.0037845371.003784537
L(12)L(\frac12) ≈\approx 1.0037845371.003784537
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)
bad2C1C_1 (1+T)2 ( 1 + T )^{2}
3 1 1
5C1C_1 (1−T)2 ( 1 - T )^{2}
good7C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
11C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
13C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
17C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
19C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
23C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
29C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
31C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
37C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
41C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
43C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + 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×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
67C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + 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×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
83C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
89C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
97C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + 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

−9.023133435735267180074635152507, −8.897890477540212442963696653360, −8.397375180912137666304748937216, −7.890029286948739667792957851102, −7.67849326775452767797140083068, −7.21853567200084208870066654996, −6.83242225115492755852226526122, −6.61138011113362149162249830833, −6.01154955992664242009573294743, −5.73557123231321272846006178503, −5.52496360970558548656805260817, −5.25536903506055819421618913054, −4.32749815004262421481440533971, −3.74303788260714063539041396806, −3.15290028292440029218077555315, −2.70217558363179904292694005913, −2.13504834424517437338992909118, −2.11926612017279930212291372171, −1.21051523781664732651206950632, −0.978875688099410759521335671866, 0.978875688099410759521335671866, 1.21051523781664732651206950632, 2.11926612017279930212291372171, 2.13504834424517437338992909118, 2.70217558363179904292694005913, 3.15290028292440029218077555315, 3.74303788260714063539041396806, 4.32749815004262421481440533971, 5.25536903506055819421618913054, 5.52496360970558548656805260817, 5.73557123231321272846006178503, 6.01154955992664242009573294743, 6.61138011113362149162249830833, 6.83242225115492755852226526122, 7.21853567200084208870066654996, 7.67849326775452767797140083068, 7.890029286948739667792957851102, 8.397375180912137666304748937216, 8.897890477540212442963696653360, 9.023133435735267180074635152507

Graph of the ZZ-function along the critical line