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Properties

Label 4-728e2-1.1-c1e2-0-2
Degree 44
Conductor 529984529984
Sign 11
Analytic cond. 33.792233.7922
Root an. cond. 2.411032.41103
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 2·9-s + 4·16-s − 10·25-s + 4·36-s + 20·43-s − 7·49-s − 8·64-s − 5·81-s + 20·100-s + 12·107-s − 36·113-s + 14·121-s + 127-s + 131-s + 137-s + 139-s − 8·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 13·169-s − 40·172-s + 173-s + 179-s + ⋯
L(s)  = 1  − 4-s − 2/3·9-s + 16-s − 2·25-s + 2/3·36-s + 3.04·43-s − 49-s − 64-s − 5/9·81-s + 2·100-s + 1.16·107-s − 3.38·113-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 169-s − 3.04·172-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

Λ(s)=(529984s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(529984s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 529984529984    =    26⋅72⋅1322^{6} \cdot 7^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 33.792233.7922
Root analytic conductor: 2.411032.41103
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 529984, ( :1/2,1/2), 1)(4,\ 529984,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.94064441430.9406444143
L(12)L(\frac12) ≈\approx 0.94064441430.9406444143
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)
bad2C2C_2 1+pT2 1 + p T^{2}
7C2C_2 1+pT2 1 + p T^{2}
13C2C_2 1+pT2 1 + p T^{2}
good3C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
5C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
23C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
29C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
41C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
43C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
59C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
61C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
67C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C22C_2^2 1−142T2+p2T4 1 - 142 T^{2} + p^{2} T^{4}
79C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
83C22C_2^2 1+158T2+p2T4 1 + 158 T^{2} + p^{2} T^{4}
89C22C_2^2 1+146T2+p2T4 1 + 146 T^{2} + p^{2} T^{4}
97C22C_2^2 1−94T2+p2T4 1 - 94 T^{2} + p^{2} 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

−8.508960193822939486478690692279, −8.045955124583760802852609303993, −7.62996356347171534713933850235, −7.38480514001354382232095887813, −6.49116829918350661385933701592, −6.12234244835669884186306396683, −5.55028250842661352481992762397, −5.41101012351539568677522043386, −4.59719182905216380872528147414, −4.14699901367412754328218294373, −3.77676223329212239240206310018, −3.05871939896826161251031442694, −2.44878784497011915880737547029, −1.59626055961315407632144667284, −0.51837991524260537333013625572, 0.51837991524260537333013625572, 1.59626055961315407632144667284, 2.44878784497011915880737547029, 3.05871939896826161251031442694, 3.77676223329212239240206310018, 4.14699901367412754328218294373, 4.59719182905216380872528147414, 5.41101012351539568677522043386, 5.55028250842661352481992762397, 6.12234244835669884186306396683, 6.49116829918350661385933701592, 7.38480514001354382232095887813, 7.62996356347171534713933850235, 8.045955124583760802852609303993, 8.508960193822939486478690692279

Graph of the ZZ-function along the critical line