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Properties

Label 4-2925e2-1.1-c0e2-0-0
Degree 44
Conductor 85556258555625
Sign 11
Analytic cond. 2.130912.13091
Root an. cond. 1.208201.20820
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  + 2·7-s − 16-s − 2·19-s + 2·31-s − 2·37-s + 2·49-s + 2·67-s + 2·73-s − 2·97-s − 2·109-s − 2·112-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2·7-s − 16-s − 2·19-s + 2·31-s − 2·37-s + 2·49-s + 2·67-s + 2·73-s − 2·97-s − 2·109-s − 2·112-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

Λ(s)=(8555625s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(8555625s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 85556258555625    =    34⋅54⋅1323^{4} \cdot 5^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 2.130912.13091
Root analytic conductor: 1.208201.20820
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 8555625, ( :0,0), 1)(4,\ 8555625,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5781090281.578109028
L(12)L(\frac12) ≈\approx 1.5781090281.578109028
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)
bad3 1 1
5 1 1
13C2C_2 1+T2 1 + T^{2}
good2C22C_2^2 1+T4 1 + T^{4}
7C1C_1×\timesC2C_2 (1−T)2(1+T2) ( 1 - T )^{2}( 1 + T^{2} )
11C22C_2^2 1+T4 1 + T^{4}
17C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
19C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
23C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
29C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
31C1C_1×\timesC2C_2 (1−T)2(1+T2) ( 1 - T )^{2}( 1 + T^{2} )
37C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
41C22C_2^2 1+T4 1 + T^{4}
43C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
47C22C_2^2 1+T4 1 + T^{4}
53C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
59C22C_2^2 1+T4 1 + T^{4}
61C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
67C1C_1×\timesC2C_2 (1−T)2(1+T2) ( 1 - T )^{2}( 1 + T^{2} )
71C22C_2^2 1+T4 1 + T^{4}
73C1C_1×\timesC2C_2 (1−T)2(1+T2) ( 1 - T )^{2}( 1 + T^{2} )
79C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
83C22C_2^2 1+T4 1 + T^{4}
89C22C_2^2 1+T4 1 + T^{4}
97C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 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

−8.961648947526123855276460415419, −8.578463707715024107279476466777, −8.321117685088286588720852072272, −8.096598058912461166318883789972, −7.924785992434291661239084190691, −7.05267160860198830843268447143, −6.94351350092379117722643286356, −6.56959945734339281681644154544, −6.18498535589391991572459290462, −5.49558565105280841295918485178, −5.24270426378048592321858880200, −4.85374351795357824825049766707, −4.37794120111384212584934433152, −4.24394926466181307307903669687, −3.72720249732470050320271435260, −2.96332388599962322690076830622, −2.45039883942702676141526581431, −1.87717345643110840097775915967, −1.78382365759962944775249701589, −0.797753375799117961488007663994, 0.797753375799117961488007663994, 1.78382365759962944775249701589, 1.87717345643110840097775915967, 2.45039883942702676141526581431, 2.96332388599962322690076830622, 3.72720249732470050320271435260, 4.24394926466181307307903669687, 4.37794120111384212584934433152, 4.85374351795357824825049766707, 5.24270426378048592321858880200, 5.49558565105280841295918485178, 6.18498535589391991572459290462, 6.56959945734339281681644154544, 6.94351350092379117722643286356, 7.05267160860198830843268447143, 7.924785992434291661239084190691, 8.096598058912461166318883789972, 8.321117685088286588720852072272, 8.578463707715024107279476466777, 8.961648947526123855276460415419

Graph of the ZZ-function along the critical line