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Properties

Label 4-546e2-1.1-c1e2-0-8
Degree 44
Conductor 298116298116
Sign 11
Analytic cond. 19.008119.0081
Root an. cond. 2.088022.08802
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  − 3·3-s + 4-s − 2·5-s + 7-s + 6·9-s − 3·12-s + 6·15-s + 16-s − 6·17-s − 2·20-s − 3·21-s − 7·25-s − 9·27-s + 28-s − 2·35-s + 6·36-s + 6·37-s − 10·43-s − 12·45-s + 26·47-s − 3·48-s − 6·49-s + 18·51-s − 20·59-s + 6·60-s + 6·63-s + 64-s + ⋯
L(s)  = 1  − 1.73·3-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 2·9-s − 0.866·12-s + 1.54·15-s + 1/4·16-s − 1.45·17-s − 0.447·20-s − 0.654·21-s − 7/5·25-s − 1.73·27-s + 0.188·28-s − 0.338·35-s + 36-s + 0.986·37-s − 1.52·43-s − 1.78·45-s + 3.79·47-s − 0.433·48-s − 6/7·49-s + 2.52·51-s − 2.60·59-s + 0.774·60-s + 0.755·63-s + 1/8·64-s + ⋯

Functional equation

Λ(s)=(298116s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(298116s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 298116298116    =    22⋅32⋅72⋅1322^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 19.008119.0081
Root analytic conductor: 2.088022.08802
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 298116, ( :1/2,1/2), 1)(4,\ 298116,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.58901062130.5890106213
L(12)L(\frac12) ≈\approx 0.58901062130.5890106213
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)
bad2C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
3C2C_2 1+pT+pT2 1 + p T + p T^{2}
7C2C_2 1−T+pT2 1 - T + p T^{2}
13C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good5C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
11C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
19C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
23C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
31C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
41C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
43C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
47C2C_2 (1−13T+pT2)2 ( 1 - 13 T + p T^{2} )^{2}
53C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
59C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
61C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
67C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
71C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
73C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
97C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 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

−8.905235966993525833388868947156, −8.225552760361780021957840571261, −7.62508756975103450239324980184, −7.45985560667588965978165906144, −6.92076425331888408807137767822, −6.38286220020503058850747043516, −5.88908450915949561478136129218, −5.73681320363144770957728323698, −4.71843544103535120742978284678, −4.61153453491182141362175201622, −4.05080151028369257637143221969, −3.36991235720395449589306296865, −2.34027315669446164505659313213, −1.64016380827058072854636802791, −0.49679880081187161703829014750, 0.49679880081187161703829014750, 1.64016380827058072854636802791, 2.34027315669446164505659313213, 3.36991235720395449589306296865, 4.05080151028369257637143221969, 4.61153453491182141362175201622, 4.71843544103535120742978284678, 5.73681320363144770957728323698, 5.88908450915949561478136129218, 6.38286220020503058850747043516, 6.92076425331888408807137767822, 7.45985560667588965978165906144, 7.62508756975103450239324980184, 8.225552760361780021957840571261, 8.905235966993525833388868947156

Graph of the ZZ-function along the critical line