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Properties

Label 4-504e2-1.1-c1e2-0-23
Degree 44
Conductor 254016254016
Sign 11
Analytic cond. 16.196216.1962
Root an. cond. 2.006102.00610
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  + 2·5-s − 2·7-s + 14·17-s + 2·25-s − 4·35-s + 4·37-s − 10·41-s + 12·43-s + 4·47-s − 3·49-s − 20·67-s + 4·79-s + 4·83-s + 28·85-s − 10·89-s + 2·101-s + 12·109-s − 28·119-s + 10·121-s + 10·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.755·7-s + 3.39·17-s + 2/5·25-s − 0.676·35-s + 0.657·37-s − 1.56·41-s + 1.82·43-s + 0.583·47-s − 3/7·49-s − 2.44·67-s + 0.450·79-s + 0.439·83-s + 3.03·85-s − 1.05·89-s + 0.199·101-s + 1.14·109-s − 2.56·119-s + 0.909·121-s + 0.894·125-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 + ⋯

Functional equation

Λ(s)=(254016s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(254016s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 254016254016    =    26⋅34⋅722^{6} \cdot 3^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 16.196216.1962
Root analytic conductor: 2.006102.00610
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 254016, ( :1/2,1/2), 1)(4,\ 254016,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1387283692.138728369
L(12)L(\frac12) ≈\approx 2.1387283692.138728369
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
3 1 1
7C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
good5C2C_2 (1−4T+pT2)(1+2T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
13C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
17C2C_2×\timesC2C_2 (1−8T+pT2)(1−6T+pT2) ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} )
19C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
23C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
29C22C_2^2 1+18T2+p2T4 1 + 18 T^{2} + p^{2} T^{4}
31C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
37C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
41C2C_2×\timesC2C_2 (1+4T+pT2)(1+6T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C2C_2×\timesC2C_2 (1−10T+pT2)(1−2T+pT2) ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} )
47C2C_2×\timesC2C_2 (1−8T+pT2)(1+4T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )
53C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
59C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
61C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1+8T+pT2)(1+12T+pT2) ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
73C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
79C2C_2×\timesC2C_2 (1−8T+pT2)(1+4T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )
83C2C_2×\timesC2C_2 (1−16T+pT2)(1+12T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C2C_2×\timesC2C_2 (1+2T+pT2)(1+8T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} )
97C22C_2^2 1+114T2+p2T4 1 + 114 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

−9.159134956812869679568273723662, −8.463305659030048724007518206089, −7.920265203540263934984211441912, −7.51943033712456739689546521679, −7.14936699479032740754868333143, −6.36722283838272124419966171712, −6.00971961788740877356019616363, −5.59074900788876333624744729444, −5.24308953152480817322793849162, −4.47425799401230125786402841817, −3.69705392804969212916691785128, −3.17530297564066487490438891901, −2.75960732049153766581847437261, −1.71508071193189601758216486522, −0.959093063501284743139378785566, 0.959093063501284743139378785566, 1.71508071193189601758216486522, 2.75960732049153766581847437261, 3.17530297564066487490438891901, 3.69705392804969212916691785128, 4.47425799401230125786402841817, 5.24308953152480817322793849162, 5.59074900788876333624744729444, 6.00971961788740877356019616363, 6.36722283838272124419966171712, 7.14936699479032740754868333143, 7.51943033712456739689546521679, 7.920265203540263934984211441912, 8.463305659030048724007518206089, 9.159134956812869679568273723662

Graph of the ZZ-function along the critical line