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Properties

Label 4-1216e2-1.1-c1e2-0-13
Degree 44
Conductor 14786561478656
Sign 11
Analytic cond. 94.280394.2803
Root an. cond. 3.116053.11605
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·7-s + 5·9-s + 6·17-s + 18·23-s + 10·25-s − 12·31-s + 12·41-s + 13·49-s − 30·63-s + 22·73-s + 24·79-s + 16·81-s − 16·97-s − 12·103-s − 24·113-s − 36·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 30·153-s + 157-s − 108·161-s + 163-s + ⋯
L(s)  = 1  − 2.26·7-s + 5/3·9-s + 1.45·17-s + 3.75·23-s + 2·25-s − 2.15·31-s + 1.87·41-s + 13/7·49-s − 3.77·63-s + 2.57·73-s + 2.70·79-s + 16/9·81-s − 1.62·97-s − 1.18·103-s − 2.25·113-s − 3.30·119-s + 2·121-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 + 2.42·153-s + 0.0798·157-s − 8.51·161-s + 0.0783·163-s + ⋯

Functional equation

Λ(s)=(1478656s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1478656s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 14786561478656    =    212⋅1922^{12} \cdot 19^{2}
Sign: 11
Analytic conductor: 94.280394.2803
Root analytic conductor: 3.116053.11605
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1478656, ( :1/2,1/2), 1)(4,\ 1478656,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4337387162.433738716
L(12)L(\frac12) ≈\approx 2.4337387162.433738716
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
19C2C_2 1+T2 1 + T^{2}
good3C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
5C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
7C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
11C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
13C22C_2^2 1−17T2+p2T4 1 - 17 T^{2} + p^{2} T^{4}
17C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
23C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
29C22C_2^2 1+23T2+p2T4 1 + 23 T^{2} + p^{2} T^{4}
31C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
37C22C_2^2 1−38T2+p2T4 1 - 38 T^{2} + p^{2} T^{4}
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
59C22C_2^2 1−109T2+p2T4 1 - 109 T^{2} + p^{2} T^{4}
61C22C_2^2 1−86T2+p2T4 1 - 86 T^{2} + p^{2} T^{4}
67C22C_2^2 1−109T2+p2T4 1 - 109 T^{2} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−11T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}
79C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
83C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{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

−9.661165550558986193511889780429, −9.452082351369450189442281696885, −9.338297278609692927807365341382, −9.064455935931297151545476565667, −8.312700185860270809450382307879, −7.77307587699736122115391753624, −7.16052590256910063539794378756, −7.08070584991519485848520445672, −6.69205073326629599870617058586, −6.50250727412608230979435762145, −5.55270838919019823822788404583, −5.43690883218433792305781071075, −4.79260310657974610597233093823, −4.35336587019880043428892678064, −3.46764598695459551704989318974, −3.39567998387535460812683799868, −3.01000527337709967171374865417, −2.24773209460051432385704753352, −1.09513157184347448438907192422, −0.859629915268150574693685825194, 0.859629915268150574693685825194, 1.09513157184347448438907192422, 2.24773209460051432385704753352, 3.01000527337709967171374865417, 3.39567998387535460812683799868, 3.46764598695459551704989318974, 4.35336587019880043428892678064, 4.79260310657974610597233093823, 5.43690883218433792305781071075, 5.55270838919019823822788404583, 6.50250727412608230979435762145, 6.69205073326629599870617058586, 7.08070584991519485848520445672, 7.16052590256910063539794378756, 7.77307587699736122115391753624, 8.312700185860270809450382307879, 9.064455935931297151545476565667, 9.338297278609692927807365341382, 9.452082351369450189442281696885, 9.661165550558986193511889780429

Graph of the ZZ-function along the critical line