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Properties

Label 8-162e4-1.1-c6e4-0-3
Degree 88
Conductor 688747536688747536
Sign 11
Analytic cond. 1.92921×1061.92921\times 10^{6}
Root an. cond. 6.104816.10481
Motivic weight 66
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  + 32·4-s − 778·7-s − 2.83e3·13-s − 1.22e4·19-s − 2.94e4·25-s − 2.48e4·28-s + 2.26e4·31-s + 1.88e5·37-s − 2.90e5·43-s + 3.86e5·49-s − 9.05e4·52-s + 7.00e5·61-s − 3.27e4·64-s − 2.40e5·67-s + 7.00e5·73-s − 3.92e5·76-s + 5.04e5·79-s + 2.20e6·91-s + 2.59e6·97-s − 9.42e5·100-s + 3.14e6·103-s − 1.94e6·109-s + 7.78e5·121-s + 7.25e5·124-s + 127-s + 131-s + 9.54e6·133-s + ⋯
L(s)  = 1  + 1/2·4-s − 2.26·7-s − 1.28·13-s − 1.78·19-s − 1.88·25-s − 1.13·28-s + 0.761·31-s + 3.72·37-s − 3.65·43-s + 3.28·49-s − 0.644·52-s + 3.08·61-s − 1/8·64-s − 0.800·67-s + 1.80·73-s − 0.894·76-s + 1.02·79-s + 2.92·91-s + 2.84·97-s − 0.942·100-s + 2.87·103-s − 1.50·109-s + 0.439·121-s + 0.380·124-s + 4.05·133-s + ⋯

Functional equation

Λ(s)=((24⋅316)s/2 ΓC(s)4 L(s)=( Λ(7−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}
Λ(s)=((24⋅316)s/2 ΓC(s+3)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 24⋅3162^{4} \cdot 3^{16}
Sign: 11
Analytic conductor: 1.92921×1061.92921\times 10^{6}
Root analytic conductor: 6.104816.10481
Motivic weight: 66
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 24⋅316, ( :3,3,3,3), 1)(8,\ 2^{4} \cdot 3^{16} ,\ ( \ : 3, 3, 3, 3 ),\ 1 )

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 3.1680326023.168032602
L(12)L(\frac12) ≈\approx 3.1680326023.168032602
L(4)L(4) 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−p5T2+p10T4 1 - p^{5} T^{2} + p^{10} T^{4}
3 1 1
good5C23C_2^3 1+1178p2T2+997059p4T4+1178p14T6+p24T8 1 + 1178 p^{2} T^{2} + 997059 p^{4} T^{4} + 1178 p^{14} T^{6} + p^{24} T^{8}
7C22C_2^2 (1+389T+33672T2+389p6T3+p12T4)2 ( 1 + 389 T + 33672 T^{2} + 389 p^{6} T^{3} + p^{12} T^{4} )^{2}
11C23C_2^3 1−778678T2−2532088949037T4−778678p12T6+p24T8 1 - 778678 T^{2} - 2532088949037 T^{4} - 778678 p^{12} T^{6} + p^{24} T^{8}
13C22C_2^2 (1+1415T−2824584T2+1415p6T3+p12T4)2 ( 1 + 1415 T - 2824584 T^{2} + 1415 p^{6} T^{3} + p^{12} T^{4} )^{2}
17C22C_2^2 (1−42670586T2+p12T4)2 ( 1 - 42670586 T^{2} + p^{12} T^{4} )^{2}
19C2C_2 (1+3067T+p6T2)4 ( 1 + 3067 T + p^{6} T^{2} )^{4}
23C23C_2^3 1−142126630T2−1714645476863421T4−142126630p12T6+p24T8 1 - 142126630 T^{2} - 1714645476863421 T^{4} - 142126630 p^{12} T^{6} + p^{24} T^{8}
29C23C_2^3 1+1021981970T2+690632363799611859T4+1021981970p12T6+p24T8 1 + 1021981970 T^{2} + 690632363799611859 T^{4} + 1021981970 p^{12} T^{6} + p^{24} T^{8}
31C22C_2^2 (1−11338T−758953437T2−11338p6T3+p12T4)2 ( 1 - 11338 T - 758953437 T^{2} - 11338 p^{6} T^{3} + p^{12} T^{4} )^{2}
37C2C_2 (1−47135T+p6T2)4 ( 1 - 47135 T + p^{6} T^{2} )^{4}
41C23C_2^3 1+9461839682T2+66962919867503675043T4+9461839682p12T6+p24T8 1 + 9461839682 T^{2} + 66962919867503675043 T^{4} + 9461839682 p^{12} T^{6} + p^{24} T^{8}
43C22C_2^2 (1+145118T+14737870875T2+145118p6T3+p12T4)2 ( 1 + 145118 T + 14737870875 T^{2} + 145118 p^{6} T^{3} + p^{12} T^{4} )^{2}
47C23C_2^3 1−10908185542T2+2797028709749255523T4−10908185542p12T6+p24T8 1 - 10908185542 T^{2} + 2797028709749255523 T^{4} - 10908185542 p^{12} T^{6} + p^{24} T^{8}
53C22C_2^2 (1+26326193614T2+p12T4)2 ( 1 + 26326193614 T^{2} + p^{12} T^{4} )^{2}
59C23C_2^3 1−46491888310T2+ 1 - 46491888310 T^{2} + 38 ⁣⋯ ⁣1938\!\cdots\!19T4−46491888310p12T6+p24T8 T^{4} - 46491888310 p^{12} T^{6} + p^{24} T^{8}
61C22C_2^2 (1−350305T+71193218664T2−350305p6T3+p12T4)2 ( 1 - 350305 T + 71193218664 T^{2} - 350305 p^{6} T^{3} + p^{12} T^{4} )^{2}
67C22C_2^2 (1+120341T−75976425888T2+120341p6T3+p12T4)2 ( 1 + 120341 T - 75976425888 T^{2} + 120341 p^{6} T^{3} + p^{12} T^{4} )^{2}
71C22C_2^2 (1−143908067234T2+p12T4)2 ( 1 - 143908067234 T^{2} + p^{12} T^{4} )^{2}
73C2C_2 (1−175151T+p6T2)4 ( 1 - 175151 T + p^{6} T^{2} )^{4}
79C22C_2^2 (1−252259T−179452852440T2−252259p6T3+p12T4)2 ( 1 - 252259 T - 179452852440 T^{2} - 252259 p^{6} T^{3} + p^{12} T^{4} )^{2}
83C23C_2^3 1+538899522770T2+ 1 + 538899522770 T^{2} + 18 ⁣⋯ ⁣3918\!\cdots\!39T4+538899522770p12T6+p24T8 T^{4} + 538899522770 p^{12} T^{6} + p^{24} T^{8}
89C22C_2^2 (1−369908068250T2+p12T4)2 ( 1 - 369908068250 T^{2} + p^{12} T^{4} )^{2}
97C22C_2^2 (1−1297105T+849509376096T2−1297105p6T3+p12T4)2 ( 1 - 1297105 T + 849509376096 T^{2} - 1297105 p^{6} T^{3} + p^{12} 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

−8.269889735291045762230086643138, −7.87306648449285120418418752146, −7.76536008061516214933571673696, −7.45311202219600358563901303880, −7.04708889338972959874146853795, −6.64968907469612463304456867717, −6.59109296226721769883920424313, −6.35889275018337499515132976036, −6.19352439112723331060588994263, −5.78155084453566623193421527522, −5.42230522924932487145231378923, −5.17834425596388211360321444792, −4.54805280720455047325942465188, −4.36704602584447569112308818743, −4.17198463795152821356581411196, −3.60018349874754483592120738510, −3.28166674042004454450124813041, −3.17640183575700694616864634103, −2.46044921420821258752194530509, −2.42377009916850530940366122988, −1.98862649485580996682960395134, −1.77142665621155530444265273022, −0.69574310441265592074250664980, −0.48689087649009089066923298043, −0.46420448697845816662695662212, 0.46420448697845816662695662212, 0.48689087649009089066923298043, 0.69574310441265592074250664980, 1.77142665621155530444265273022, 1.98862649485580996682960395134, 2.42377009916850530940366122988, 2.46044921420821258752194530509, 3.17640183575700694616864634103, 3.28166674042004454450124813041, 3.60018349874754483592120738510, 4.17198463795152821356581411196, 4.36704602584447569112308818743, 4.54805280720455047325942465188, 5.17834425596388211360321444792, 5.42230522924932487145231378923, 5.78155084453566623193421527522, 6.19352439112723331060588994263, 6.35889275018337499515132976036, 6.59109296226721769883920424313, 6.64968907469612463304456867717, 7.04708889338972959874146853795, 7.45311202219600358563901303880, 7.76536008061516214933571673696, 7.87306648449285120418418752146, 8.269889735291045762230086643138

Graph of the ZZ-function along the critical line