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Properties

Label 8-416e4-1.1-c1e4-0-9
Degree 88
Conductor 2994837913629948379136
Sign 11
Analytic cond. 121.753121.753
Root an. cond. 1.822571.82257
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  + 3·9-s − 4·13-s + 14·17-s − 4·25-s + 10·29-s + 18·37-s − 6·41-s − 5·49-s + 16·53-s − 6·61-s + 9·81-s + 6·89-s + 30·97-s − 34·101-s − 10·113-s − 12·117-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 42·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 9-s − 1.10·13-s + 3.39·17-s − 4/5·25-s + 1.85·29-s + 2.95·37-s − 0.937·41-s − 5/7·49-s + 2.19·53-s − 0.768·61-s + 81-s + 0.635·89-s + 3.04·97-s − 3.38·101-s − 0.940·113-s − 1.10·117-s + 3/11·121-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 + 3.39·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

Λ(s)=((220⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((220⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \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: 220⋅1342^{20} \cdot 13^{4}
Sign: 11
Analytic conductor: 121.753121.753
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 220⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{20} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 3.0537969563.053796956
L(12)L(\frac12) ≈\approx 3.0537969563.053796956
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
13C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
good3C2C_2×\timesC22C_2^2 (1−pT2)2(1+pT2+p2T4) ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} )
5C22C_2^2 (1+2T2+p2T4)2 ( 1 + 2 T^{2} + p^{2} T^{4} )^{2}
7C23C_2^3 1+5T2−24T4+5p2T6+p4T8 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8}
11C23C_2^3 1−3T2−112T4−3p2T6+p4T8 1 - 3 T^{2} - 112 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8}
17C22C_2^2 (1−7T+32T2−7pT3+p2T4)2 ( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2}
19C23C_2^3 1+13T2−192T4+13p2T6+p4T8 1 + 13 T^{2} - 192 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8}
23C23C_2^3 1−19T2−168T4−19p2T6+p4T8 1 - 19 T^{2} - 168 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8}
29C22C_2^2 (1−5T−4T2−5pT3+p2T4)2 ( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2}
31C22C_2^2 (1−58T2+p2T4)2 ( 1 - 58 T^{2} + p^{2} T^{4} )^{2}
37C2C_2 (1−10T+pT2)2(1+T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2}
41C22C_2^2 (1+3T+44T2+3pT3+p2T4)2 ( 1 + 3 T + 44 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}
43C23C_2^3 1−59T2+1632T4−59p2T6+p4T8 1 - 59 T^{2} + 1632 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8}
47C22C_2^2 (1−78T2+p2T4)2 ( 1 - 78 T^{2} + p^{2} T^{4} )^{2}
53C2C_2 (1−4T+pT2)4 ( 1 - 4 T + p T^{2} )^{4}
59C23C_2^3 1+69T2+1280T4+69p2T6+p4T8 1 + 69 T^{2} + 1280 T^{4} + 69 p^{2} T^{6} + p^{4} T^{8}
61C22C_2^2 (1+3T−52T2+3pT3+p2T4)2 ( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}
67C23C_2^3 1+125T2+11136T4+125p2T6+p4T8 1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8}
71C23C_2^3 1+93T2+3608T4+93p2T6+p4T8 1 + 93 T^{2} + 3608 T^{4} + 93 p^{2} T^{6} + p^{4} T^{8}
73C22C_2^2 (1−134T2+p2T4)2 ( 1 - 134 T^{2} + p^{2} T^{4} )^{2}
79C22C_2^2 (1+146T2+p2T4)2 ( 1 + 146 T^{2} + p^{2} T^{4} )^{2}
83C22C_2^2 (1+30T2+p2T4)2 ( 1 + 30 T^{2} + p^{2} T^{4} )^{2}
89C22C_2^2 (1−3T+92T2−3pT3+p2T4)2 ( 1 - 3 T + 92 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}
97C22C_2^2 (1−15T+172T2−15pT3+p2T4)2 ( 1 - 15 T + 172 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2}
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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

−7.85999433137825841238346958449, −7.84837922966356476336148971811, −7.67431865783985467018022019400, −7.65184186003859798424435421020, −7.09480649625657389865806807161, −7.00133026286185184359867541163, −6.55885706918457310138539892417, −6.43284603740802420437548879113, −6.18570872055557231223196832515, −5.69139142263664641777595746280, −5.51169308262128402487868568562, −5.50461241793743242696247178041, −4.99253488121029589828855222896, −4.69551890972061568647975851003, −4.55274025705988453260866714081, −4.19300724258586834100372526043, −3.77900151792917559691653252644, −3.64303988130129048774757290899, −3.13280708837345839580850573153, −2.93104039303353373560077906867, −2.54894164571844234500184399434, −2.20001653520271263739376382071, −1.57833498637778882692703617024, −1.11471873287324465725549016299, −0.797184005518751079591302016406, 0.797184005518751079591302016406, 1.11471873287324465725549016299, 1.57833498637778882692703617024, 2.20001653520271263739376382071, 2.54894164571844234500184399434, 2.93104039303353373560077906867, 3.13280708837345839580850573153, 3.64303988130129048774757290899, 3.77900151792917559691653252644, 4.19300724258586834100372526043, 4.55274025705988453260866714081, 4.69551890972061568647975851003, 4.99253488121029589828855222896, 5.50461241793743242696247178041, 5.51169308262128402487868568562, 5.69139142263664641777595746280, 6.18570872055557231223196832515, 6.43284603740802420437548879113, 6.55885706918457310138539892417, 7.00133026286185184359867541163, 7.09480649625657389865806807161, 7.65184186003859798424435421020, 7.67431865783985467018022019400, 7.84837922966356476336148971811, 7.85999433137825841238346958449

Graph of the ZZ-function along the critical line