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Properties

Label 4-3e10-1.1-c1e2-0-4
Degree 44
Conductor 5904959049
Sign 11
Analytic cond. 3.765013.76501
Root an. cond. 1.392961.39296
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  + 2·4-s + 4·7-s + 7·13-s − 2·19-s + 5·25-s + 8·28-s − 11·31-s − 20·37-s − 5·43-s + 7·49-s + 14·52-s + 61-s − 8·64-s − 5·67-s − 14·73-s − 4·76-s + 13·79-s + 28·91-s − 5·97-s + 10·100-s + 13·103-s − 38·109-s + 11·121-s − 22·124-s + 127-s + 131-s − 8·133-s + ⋯
L(s)  = 1  + 4-s + 1.51·7-s + 1.94·13-s − 0.458·19-s + 25-s + 1.51·28-s − 1.97·31-s − 3.28·37-s − 0.762·43-s + 49-s + 1.94·52-s + 0.128·61-s − 64-s − 0.610·67-s − 1.63·73-s − 0.458·76-s + 1.46·79-s + 2.93·91-s − 0.507·97-s + 100-s + 1.28·103-s − 3.63·109-s + 121-s − 1.97·124-s + 0.0887·127-s + 0.0873·131-s − 0.693·133-s + ⋯

Functional equation

Λ(s)=(59049s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(59049s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5904959049    =    3103^{10}
Sign: 11
Analytic conductor: 3.765013.76501
Root analytic conductor: 1.392961.39296
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 59049, ( :1/2,1/2), 1)(4,\ 59049,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1639891862.163989186
L(12)L(\frac12) ≈\approx 2.1639891862.163989186
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)
bad3 1 1
good2C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
5C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
7C2C_2 (1−5T+pT2)(1+T+pT2) ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} )
11C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
13C2C_2 (1−5T+pT2)(1−2T+pT2) ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} )
17C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
19C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
23C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
29C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
31C2C_2 (1+4T+pT2)(1+7T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} )
37C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
41C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
43C2C_2 (1−8T+pT2)(1+13T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} )
47C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
53C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
59C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
61C2C_2 (1−14T+pT2)(1+13T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} )
67C2C_2 (1−11T+pT2)(1+16T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} )
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
79C2C_2 (1−17T+pT2)(1+4T+pT2) ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} )
83C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97C2C_2 (1−14T+pT2)(1+19T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 19 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

−11.97176877688928411767925748799, −11.96324524829119598033844710065, −11.24056823535085208879544823194, −10.96096379134801597032705417011, −10.56037986866456853977201504282, −10.41906286223794603794670020728, −9.088340175494034891205275362919, −9.032681696819044858699813405982, −8.305379320250581603205108826742, −8.123564695784635754761888175223, −7.20965693379886886712998282992, −6.94654208274578141535972637128, −6.39529797301599778835698702081, −5.61645945366736046919660453916, −5.25867214836061762996343017701, −4.47899069305727190389317624933, −3.72171311063105546675373390782, −3.10820885202341447541150209339, −1.83508433814506093189173428511, −1.61416762214392151984078775814, 1.61416762214392151984078775814, 1.83508433814506093189173428511, 3.10820885202341447541150209339, 3.72171311063105546675373390782, 4.47899069305727190389317624933, 5.25867214836061762996343017701, 5.61645945366736046919660453916, 6.39529797301599778835698702081, 6.94654208274578141535972637128, 7.20965693379886886712998282992, 8.123564695784635754761888175223, 8.305379320250581603205108826742, 9.032681696819044858699813405982, 9.088340175494034891205275362919, 10.41906286223794603794670020728, 10.56037986866456853977201504282, 10.96096379134801597032705417011, 11.24056823535085208879544823194, 11.96324524829119598033844710065, 11.97176877688928411767925748799

Graph of the ZZ-function along the critical line