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Properties

Label 4-320e2-1.1-c1e2-0-0
Degree 44
Conductor 102400102400
Sign 11
Analytic cond. 6.529116.52911
Root an. cond. 1.598501.59850
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  − 4·5-s + 10·13-s − 10·17-s + 11·25-s − 10·37-s + 16·41-s − 10·53-s + 24·61-s − 40·65-s + 10·73-s − 9·81-s + 40·85-s + 10·97-s − 4·101-s + 30·113-s + 22·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 50·169-s + ⋯
L(s)  = 1  − 1.78·5-s + 2.77·13-s − 2.42·17-s + 11/5·25-s − 1.64·37-s + 2.49·41-s − 1.37·53-s + 3.07·61-s − 4.96·65-s + 1.17·73-s − 81-s + 4.33·85-s + 1.01·97-s − 0.398·101-s + 2.82·113-s + 2·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.84·169-s + ⋯

Functional equation

Λ(s)=(102400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(102400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 102400102400    =    212⋅522^{12} \cdot 5^{2}
Sign: 11
Analytic conductor: 6.529116.52911
Root analytic conductor: 1.598501.59850
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 102400, ( :1/2,1/2), 1)(4,\ 102400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0280800731.028080073
L(12)L(\frac12) ≈\approx 1.0280800731.028080073
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)
bad2 1 1
5C2C_2 1+4T+pT2 1 + 4 T + p T^{2}
good3C22C_2^2 1+p2T4 1 + p^{2} T^{4}
7C22C_2^2 1+p2T4 1 + p^{2} T^{4}
11C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
13C2C_2 (1−6T+pT2)(1−4T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} )
17C2C_2 (1+2T+pT2)(1+8T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C22C_2^2 1+p2T4 1 + p^{2} T^{4}
29C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
31C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
37C2C_2 (1−2T+pT2)(1+12T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
43C22C_2^2 1+p2T4 1 + p^{2} T^{4}
47C22C_2^2 1+p2T4 1 + p^{2} T^{4}
53C2C_2 (1−4T+pT2)(1+14T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} )
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
67C22C_2^2 1+p2T4 1 + p^{2} T^{4}
71C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
73C2C_2 (1−16T+pT2)(1+6T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C22C_2^2 1+p2T4 1 + p^{2} T^{4}
89C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
97C2C_2 (1−18T+pT2)(1+8T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 8 T + p 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

−11.45823637494045206387330414327, −11.34802083647064143476429057640, −11.17739061108679690381095781286, −10.75088767626022843866068533848, −10.21521912329344702226891179652, −9.290739722645913941443564781581, −8.753983348100711241549479155291, −8.674534544629272867030782860446, −8.203325745555624988059499181647, −7.65282297564612043595752202219, −7.00036321406455903231163002005, −6.59336922864393581770584593663, −6.17080292838263670808391925213, −5.40503843578542021326014796610, −4.55279357045582279894781279974, −4.12396539936214544141183389183, −3.73628229388653838275156087841, −3.12558842572288379734781975579, −2.02547983030528401194618204137, −0.76085770125354409810161120625, 0.76085770125354409810161120625, 2.02547983030528401194618204137, 3.12558842572288379734781975579, 3.73628229388653838275156087841, 4.12396539936214544141183389183, 4.55279357045582279894781279974, 5.40503843578542021326014796610, 6.17080292838263670808391925213, 6.59336922864393581770584593663, 7.00036321406455903231163002005, 7.65282297564612043595752202219, 8.203325745555624988059499181647, 8.674534544629272867030782860446, 8.753983348100711241549479155291, 9.290739722645913941443564781581, 10.21521912329344702226891179652, 10.75088767626022843866068533848, 11.17739061108679690381095781286, 11.34802083647064143476429057640, 11.45823637494045206387330414327

Graph of the ZZ-function along the critical line