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Properties

Label 8-650e4-1.1-c1e4-0-10
Degree 88
Conductor 178506250000178506250000
Sign 11
Analytic cond. 725.707725.707
Root an. cond. 2.278212.27821
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 6·9-s − 8·11-s − 2·29-s + 16·31-s − 6·36-s + 18·41-s − 8·44-s + 2·49-s − 8·59-s − 14·61-s − 64-s + 16·71-s + 16·79-s + 9·81-s − 12·89-s + 48·99-s − 14·101-s + 8·109-s − 2·116-s + 38·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  + 1/2·4-s − 2·9-s − 2.41·11-s − 0.371·29-s + 2.87·31-s − 36-s + 2.81·41-s − 1.20·44-s + 2/7·49-s − 1.04·59-s − 1.79·61-s − 1/8·64-s + 1.89·71-s + 1.80·79-s + 81-s − 1.27·89-s + 4.82·99-s − 1.39·101-s + 0.766·109-s − 0.185·116-s + 3.45·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯

Functional equation

Λ(s)=((24⋅58⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((24⋅58⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 24⋅58⋅1342^{4} \cdot 5^{8} \cdot 13^{4}
Sign: 11
Analytic conductor: 725.707725.707
Root analytic conductor: 2.278212.27821
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 24⋅58⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{4} \cdot 5^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.5874857941.587485794
L(12)L(\frac12) ≈\approx 1.5874857941.587485794
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)
bad2C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
5 1 1
13C22C_2^2 1+23T2+p2T4 1 + 23 T^{2} + p^{2} T^{4}
good3C22C_2^2 (1+pT2+p2T4)2 ( 1 + p T^{2} + p^{2} T^{4} )^{2}
7C22C_2^2×\timesC22C_2^2 (1−13T2+p2T4)(1+11T2+p2T4) ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} )
11C22C_2^2 (1+4T+5T2+4pT3+p2T4)2 ( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
17C23C_2^3 1+25T2+336T4+25p2T6+p4T8 1 + 25 T^{2} + 336 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8}
19C22C_2^2 (1−pT2+p2T4)2 ( 1 - p T^{2} + p^{2} T^{4} )^{2}
23C23C_2^3 1+30T2+371T4+30p2T6+p4T8 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8}
29C22C_2^2 (1+T−28T2+pT3+p2T4)2 ( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} )^{2}
31C2C_2 (1−4T+pT2)4 ( 1 - 4 T + p T^{2} )^{4}
37C23C_2^3 1+65T2+2856T4+65p2T6+p4T8 1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8}
41C22C_2^2 (1−9T+40T2−9pT3+p2T4)2 ( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2}
43C22C_2^2×\timesC22C_2^2 (1−61T2+p2T4)(1+83T2+p2T4) ( 1 - 61 T^{2} + p^{2} T^{4} )( 1 + 83 T^{2} + p^{2} T^{4} )
47C22C_2^2 (1−30T2+p2T4)2 ( 1 - 30 T^{2} + p^{2} T^{4} )^{2}
53C22C_2^2 (1−25T2+p2T4)2 ( 1 - 25 T^{2} + p^{2} T^{4} )^{2}
59C22C_2^2 (1+4T−43T2+4pT3+p2T4)2 ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
61C22C_2^2 (1+7T−12T2+7pT3+p2T4)2 ( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2}
67C23C_2^3 1+118T2+9435T4+118p2T6+p4T8 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8}
71C22C_2^2 (1−8T−7T2−8pT3+p2T4)2 ( 1 - 8 T - 7 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
73C22C_2^2 (1−25T2+p2T4)2 ( 1 - 25 T^{2} + p^{2} T^{4} )^{2}
79C2C_2 (1−4T+pT2)4 ( 1 - 4 T + p T^{2} )^{4}
83C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
89C22C_2^2 (1+6T−53T2+6pT3+p2T4)2 ( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
97C23C_2^3 1+190T2+26691T4+190p2T6+p4T8 1 + 190 T^{2} + 26691 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.64440090888471205407183644984, −7.47057953950097738401686145619, −7.17761209985030149240914205021, −6.96572641400873621483563268841, −6.51742077064137406483133723657, −6.43385728576875241311207717413, −6.12461220682501026577217382826, −5.79446905463061659546830680489, −5.60034404424060112891284323826, −5.56532112037077169937601262187, −5.51483680838494261696577765261, −4.71423993774995886462195233697, −4.60682463175509487754318673477, −4.51518566006225694065999702056, −4.46428528797228893896671213373, −3.59453650167956852239802233846, −3.31839395665777794749042032370, −3.27107762032616699035558959007, −2.73724946181445990725714237399, −2.68037382483886837285955979260, −2.33821431991323016906059493486, −2.23814912793082852776572003796, −1.57556347262722484911703511943, −0.78846958689503377595506759044, −0.46029968845804465031127256987, 0.46029968845804465031127256987, 0.78846958689503377595506759044, 1.57556347262722484911703511943, 2.23814912793082852776572003796, 2.33821431991323016906059493486, 2.68037382483886837285955979260, 2.73724946181445990725714237399, 3.27107762032616699035558959007, 3.31839395665777794749042032370, 3.59453650167956852239802233846, 4.46428528797228893896671213373, 4.51518566006225694065999702056, 4.60682463175509487754318673477, 4.71423993774995886462195233697, 5.51483680838494261696577765261, 5.56532112037077169937601262187, 5.60034404424060112891284323826, 5.79446905463061659546830680489, 6.12461220682501026577217382826, 6.43385728576875241311207717413, 6.51742077064137406483133723657, 6.96572641400873621483563268841, 7.17761209985030149240914205021, 7.47057953950097738401686145619, 7.64440090888471205407183644984

Graph of the ZZ-function along the critical line