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Properties

Label 4-665e2-1.1-c0e2-0-2
Degree 44
Conductor 442225442225
Sign 11
Analytic cond. 0.1101430.110143
Root an. cond. 0.5760880.576088
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  + 2·5-s − 16-s + 2·17-s − 2·19-s − 2·23-s + 3·25-s + 2·43-s − 2·47-s − 49-s − 2·73-s − 2·80-s − 81-s − 2·83-s + 4·85-s − 4·95-s − 4·115-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 2·5-s − 16-s + 2·17-s − 2·19-s − 2·23-s + 3·25-s + 2·43-s − 2·47-s − 49-s − 2·73-s − 2·80-s − 81-s − 2·83-s + 4·85-s − 4·95-s − 4·115-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯

Functional equation

Λ(s)=(442225s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 442225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(442225s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 442225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 442225442225    =    52⋅72⋅1925^{2} \cdot 7^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 0.1101430.110143
Root analytic conductor: 0.5760880.576088
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 442225, ( :0,0), 1)(4,\ 442225,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1470992421.147099242
L(12)L(\frac12) ≈\approx 1.1470992421.147099242
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)
bad5C1C_1 (1−T)2 ( 1 - T )^{2}
7C2C_2 1+T2 1 + T^{2}
19C1C_1 (1+T)2 ( 1 + T )^{2}
good2C22C_2^2 1+T4 1 + T^{4}
3C22C_2^2 1+T4 1 + T^{4}
11C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
13C22C_2^2 1+T4 1 + T^{4}
17C1C_1×\timesC2C_2 (1−T)2(1+T2) ( 1 - T )^{2}( 1 + T^{2} )
23C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
29C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
31C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
37C22C_2^2 1+T4 1 + T^{4}
41C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
43C1C_1×\timesC2C_2 (1−T)2(1+T2) ( 1 - T )^{2}( 1 + T^{2} )
47C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
53C22C_2^2 1+T4 1 + T^{4}
59C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
61C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
67C22C_2^2 1+T4 1 + T^{4}
71C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
73C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
79C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
83C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
89C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
97C22C_2^2 1+T4 1 + 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

−10.72615927969041585731995588934, −10.32918905788299336334882037230, −9.988376377710077954788629609076, −9.924446417475096846171340070518, −9.138144032372668070630783235816, −9.030199195551708318378212193397, −8.378705614004823319141208676483, −7.975188589667517137895589762436, −7.48599048248338289994439141166, −6.68978854237765795045860714910, −6.47165642711306764012317447542, −6.03041039182173119252884507528, −5.57683985062434647262916555687, −5.28188730254292165510384571542, −4.30696940431278640308083674493, −4.26811540361817901088843956979, −3.12721838566535812553793476446, −2.65469169745686763492984916862, −1.91808551249598145800203958344, −1.54435994057161357153769802834, 1.54435994057161357153769802834, 1.91808551249598145800203958344, 2.65469169745686763492984916862, 3.12721838566535812553793476446, 4.26811540361817901088843956979, 4.30696940431278640308083674493, 5.28188730254292165510384571542, 5.57683985062434647262916555687, 6.03041039182173119252884507528, 6.47165642711306764012317447542, 6.68978854237765795045860714910, 7.48599048248338289994439141166, 7.975188589667517137895589762436, 8.378705614004823319141208676483, 9.030199195551708318378212193397, 9.138144032372668070630783235816, 9.924446417475096846171340070518, 9.988376377710077954788629609076, 10.32918905788299336334882037230, 10.72615927969041585731995588934

Graph of the ZZ-function along the critical line