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Properties

Label 8-2160e4-1.1-c2e4-0-10
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 124·19-s − 40·25-s + 64·31-s + 178·49-s − 4·61-s + 4·79-s + 416·109-s + 304·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 206·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 6.52·19-s − 8/5·25-s + 2.06·31-s + 3.63·49-s − 0.0655·61-s + 4/79·79-s + 3.81·109-s + 2.51·121-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 − 1.21·169-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 + 0.00448·223-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 13.1801417013.18014170
L(12)L(\frac12) ≈\approx 13.1801417013.18014170
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
5C22C_2^2 1+8pT2+p4T4 1 + 8 p T^{2} + p^{4} T^{4}
good7C22C_2^2 (1−89T2+p4T4)2 ( 1 - 89 T^{2} + p^{4} T^{4} )^{2}
11C22C_2^2 (1−152T2+p4T4)2 ( 1 - 152 T^{2} + p^{4} T^{4} )^{2}
13C22C_2^2 (1+103T2+p4T4)2 ( 1 + 103 T^{2} + p^{4} T^{4} )^{2}
17C22C_2^2 (1+418T2+p4T4)2 ( 1 + 418 T^{2} + p^{4} T^{4} )^{2}
19C2C_2 (1−31T+p2T2)4 ( 1 - 31 T + p^{2} T^{2} )^{4}
23C22C_2^2 (1+568T2+p4T4)2 ( 1 + 568 T^{2} + p^{4} T^{4} )^{2}
29C22C_2^2 (1+568T2+p4T4)2 ( 1 + 568 T^{2} + p^{4} T^{4} )^{2}
31C2C_2 (1−16T+p2T2)4 ( 1 - 16 T + p^{2} T^{2} )^{4}
37C22C_2^2 (1−2009T2+p4T4)2 ( 1 - 2009 T^{2} + p^{4} T^{4} )^{2}
41C22C_2^2 (1−1112T2+p4T4)2 ( 1 - 1112 T^{2} + p^{4} T^{4} )^{2}
43C22C_2^2 (1−1394T2+p4T4)2 ( 1 - 1394 T^{2} + p^{4} T^{4} )^{2}
47C22C_2^2 (1+4258T2+p4T4)2 ( 1 + 4258 T^{2} + p^{4} T^{4} )^{2}
53C22C_2^2 (1+3928T2+p4T4)2 ( 1 + 3928 T^{2} + p^{4} T^{4} )^{2}
59C22C_2^2 (1−5522T2+p4T4)2 ( 1 - 5522 T^{2} + p^{4} T^{4} )^{2}
61C2C_2 (1+T+p2T2)4 ( 1 + T + p^{2} T^{2} )^{4}
67C22C_2^2 (1−8537T2+p4T4)2 ( 1 - 8537 T^{2} + p^{4} T^{4} )^{2}
71C22C_2^2 (1−9272T2+p4T4)2 ( 1 - 9272 T^{2} + p^{4} T^{4} )^{2}
73C22C_2^2 (1−9929T2+p4T4)2 ( 1 - 9929 T^{2} + p^{4} T^{4} )^{2}
79C2C_2 (1−T+p2T2)4 ( 1 - T + p^{2} T^{2} )^{4}
83C22C_2^2 (1+1528T2+p4T4)2 ( 1 + 1528 T^{2} + p^{4} T^{4} )^{2}
89C22C_2^2 (1−2882T2+p4T4)2 ( 1 - 2882 T^{2} + p^{4} T^{4} )^{2}
97C22C_2^2 (1−10169T2+p4T4)2 ( 1 - 10169 T^{2} + p^{4} 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

−6.21438062759510482786683276719, −5.94708946174985058045013498214, −5.81473184084430859636675450310, −5.61462112950724982786437976544, −5.40048490831536428858916961246, −5.21606256385350892610978870699, −5.16241608849150930528975852097, −4.83036610195798083841593859173, −4.53155380859506497815264655145, −4.42180453147960649836156957675, −4.08253635947917157952260224222, −3.75819838196827559920334672383, −3.58020344766317732482899897263, −3.38837375895475496290333927180, −3.18242905278834579060466774321, −2.94895449791701500892313100055, −2.72853579326341594798068331471, −2.52404370638662269982798470188, −2.12437926927513239643605328073, −1.78980819500163831422210000937, −1.48630706301953191401225376768, −1.07858715357387718607577657506, −0.822736455912634815373562912243, −0.817008663883462826980026789393, −0.45428692620403345115813374308, 0.45428692620403345115813374308, 0.817008663883462826980026789393, 0.822736455912634815373562912243, 1.07858715357387718607577657506, 1.48630706301953191401225376768, 1.78980819500163831422210000937, 2.12437926927513239643605328073, 2.52404370638662269982798470188, 2.72853579326341594798068331471, 2.94895449791701500892313100055, 3.18242905278834579060466774321, 3.38837375895475496290333927180, 3.58020344766317732482899897263, 3.75819838196827559920334672383, 4.08253635947917157952260224222, 4.42180453147960649836156957675, 4.53155380859506497815264655145, 4.83036610195798083841593859173, 5.16241608849150930528975852097, 5.21606256385350892610978870699, 5.40048490831536428858916961246, 5.61462112950724982786437976544, 5.81473184084430859636675450310, 5.94708946174985058045013498214, 6.21438062759510482786683276719

Graph of the ZZ-function along the critical line