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Properties

Label 8-2160e4-1.1-c2e4-0-5
Degree 88
Conductor 2.177×10132.177\times 10^{13}
Sign 11
Analytic cond. 1.19992×1071.19992\times 10^{7}
Root an. cond. 7.671747.67174
Motivic weight 22
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  − 32·13-s + 10·25-s + 8·37-s + 196·49-s − 124·61-s − 296·73-s − 128·97-s + 388·109-s + 268·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 36·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 2.46·13-s + 2/5·25-s + 8/37·37-s + 4·49-s − 2.03·61-s − 4.05·73-s − 1.31·97-s + 3.55·109-s + 2.21·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.213·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯

Functional equation

Λ(s)=((216⋅312⋅54)s/2 ΓC(s)4 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((216⋅312⋅54)s/2 ΓC(s+1)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅312⋅542^{16} \cdot 3^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 1.19992×1071.19992\times 10^{7}
Root analytic conductor: 7.671747.67174
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅312⋅54, ( :1,1,1,1), 1)(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.1580219222.158021922
L(12)L(\frac12) ≈\approx 2.1580219222.158021922
L(2)L(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
5C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
good7C1C_1×\timesC1C_1 (1−pT)4(1+pT)4 ( 1 - p T )^{4}( 1 + p T )^{4}
11C22C_2^2 (1−134T2+p4T4)2 ( 1 - 134 T^{2} + p^{4} T^{4} )^{2}
13C2C_2 (1+8T+p2T2)4 ( 1 + 8 T + p^{2} T^{2} )^{4}
17C22C_2^2 (1+533T2+p4T4)2 ( 1 + 533 T^{2} + p^{4} T^{4} )^{2}
19C22C_2^2 (1−587T2+p4T4)2 ( 1 - 587 T^{2} + p^{4} T^{4} )^{2}
23C22C_2^2 (1−1031T2+p4T4)2 ( 1 - 1031 T^{2} + p^{4} T^{4} )^{2}
29C22C_2^2 (1+962T2+p4T4)2 ( 1 + 962 T^{2} + p^{4} T^{4} )^{2}
31C22C_2^2 (1−707T2+p4T4)2 ( 1 - 707 T^{2} + p^{4} T^{4} )^{2}
37C2C_2 (1−2T+p2T2)4 ( 1 - 2 T + p^{2} T^{2} )^{4}
41C22C_2^2 (1+1742T2+p4T4)2 ( 1 + 1742 T^{2} + p^{4} T^{4} )^{2}
43C22C_2^2 (1−3158T2+p4T4)2 ( 1 - 3158 T^{2} + p^{4} T^{4} )^{2}
47C22C_2^2 (1−3986T2+p4T4)2 ( 1 - 3986 T^{2} + p^{4} T^{4} )^{2}
53C22C_2^2 (1+5213T2+p4T4)2 ( 1 + 5213 T^{2} + p^{4} T^{4} )^{2}
59C22C_2^2 (1−5234T2+p4T4)2 ( 1 - 5234 T^{2} + p^{4} T^{4} )^{2}
61C2C_2 (1+31T+p2T2)4 ( 1 + 31 T + p^{2} T^{2} )^{4}
67C22C_2^2 (1−4118T2+p4T4)2 ( 1 - 4118 T^{2} + p^{4} T^{4} )^{2}
71C22C_2^2 (1−9974T2+p4T4)2 ( 1 - 9974 T^{2} + p^{4} T^{4} )^{2}
73C2C_2 (1+74T+p2T2)4 ( 1 + 74 T + p^{2} T^{2} )^{4}
79C22C_2^2 (1−1547T2+p4T4)2 ( 1 - 1547 T^{2} + p^{4} T^{4} )^{2}
83C22C_2^2 (1−7703T2+p4T4)2 ( 1 - 7703 T^{2} + p^{4} T^{4} )^{2}
89C22C_2^2 (1+11342T2+p4T4)2 ( 1 + 11342 T^{2} + p^{4} T^{4} )^{2}
97C2C_2 (1+32T+p2T2)4 ( 1 + 32 T + p^{2} T^{2} )^{4}
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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

−6.21895743555541327273323448301, −5.94145548463962808516832303388, −5.89927398324106580499741039614, −5.62517647176238080416208205804, −5.51245406969399305721300450409, −5.10895319406599392552444750615, −4.99257406280318301221567393053, −4.75198028039488891325858033454, −4.51717659651994571545481954558, −4.43699503185256766995682961944, −4.18981099596546276331650643148, −3.94377149640341345216257786874, −3.67479878854967632914320046839, −3.29334637581889975181995960688, −3.13028483026825790248267859379, −2.79940763442211789076747369219, −2.70864148126329691178979581925, −2.36514137020532690840035136381, −2.28264846135324403622180586183, −1.77987761084753340742937143830, −1.70625317469290917021147609729, −1.16017625810997476764188873012, −0.937447228610896350048525463048, −0.42948900672542783707939569905, −0.25796796134412198210283112372, 0.25796796134412198210283112372, 0.42948900672542783707939569905, 0.937447228610896350048525463048, 1.16017625810997476764188873012, 1.70625317469290917021147609729, 1.77987761084753340742937143830, 2.28264846135324403622180586183, 2.36514137020532690840035136381, 2.70864148126329691178979581925, 2.79940763442211789076747369219, 3.13028483026825790248267859379, 3.29334637581889975181995960688, 3.67479878854967632914320046839, 3.94377149640341345216257786874, 4.18981099596546276331650643148, 4.43699503185256766995682961944, 4.51717659651994571545481954558, 4.75198028039488891325858033454, 4.99257406280318301221567393053, 5.10895319406599392552444750615, 5.51245406969399305721300450409, 5.62517647176238080416208205804, 5.89927398324106580499741039614, 5.94145548463962808516832303388, 6.21895743555541327273323448301

Graph of the ZZ-function along the critical line