Properties

Label 8-624e4-1.1-c5e4-0-0
Degree 88
Conductor 151613669376151613669376
Sign 11
Analytic cond. 1.00318×1081.00318\times 10^{8}
Root an. cond. 10.003910.0039
Motivic weight 55
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 36·3-s + 810·9-s + 312·13-s + 384·17-s − 2.42e3·23-s + 6.62e3·25-s − 1.45e4·27-s − 3.96e3·29-s − 1.12e4·39-s − 4.83e3·43-s + 4.91e4·49-s − 1.38e4·51-s − 1.82e3·53-s − 1.00e5·61-s + 8.72e4·69-s − 2.38e5·75-s + 2.05e5·79-s + 2.29e5·81-s + 1.42e5·87-s + 1.09e5·101-s + 8.31e4·103-s − 5.00e5·107-s − 3.44e5·113-s + 2.52e5·117-s + 6.17e5·121-s + 127-s + 1.73e5·129-s + ⋯
L(s)  = 1  − 2.30·3-s + 10/3·9-s + 0.512·13-s + 0.322·17-s − 0.955·23-s + 2.11·25-s − 3.84·27-s − 0.874·29-s − 1.18·39-s − 0.398·43-s + 2.92·49-s − 0.744·51-s − 0.0891·53-s − 3.47·61-s + 2.20·69-s − 4.89·75-s + 3.70·79-s + 35/9·81-s + 2.01·87-s + 1.06·101-s + 0.772·103-s − 4.22·107-s − 2.53·113-s + 1.70·117-s + 3.83·121-s + 5.50e−6·127-s + 0.920·129-s + ⋯

Functional equation

Λ(s)=((21634134)s/2ΓC(s)4L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
Λ(s)=((21634134)s/2ΓC(s+5/2)4L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216341342^{16} \cdot 3^{4} \cdot 13^{4}
Sign: 11
Analytic conductor: 1.00318×1081.00318\times 10^{8}
Root analytic conductor: 10.003910.0039
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 21634134, ( :5/2,5/2,5/2,5/2), 1)(8,\ 2^{16} \cdot 3^{4} \cdot 13^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )

Particular Values

L(3)L(3) \approx 0.48238599600.4823859960
L(12)L(\frac12) \approx 0.48238599600.4823859960
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3C1C_1 (1+p2T)4 ( 1 + p^{2} T )^{4}
13D4D_{4} 124pT146p3T224p6T3+p10T4 1 - 24 p T - 146 p^{3} T^{2} - 24 p^{6} T^{3} + p^{10} T^{4}
good5C22C2C_2^2 \wr C_2 16624T2+1083838p2T46624p10T6+p20T8 1 - 6624 T^{2} + 1083838 p^{2} T^{4} - 6624 p^{10} T^{6} + p^{20} T^{8}
7C22C2C_2^2 \wr C_2 149132T2+1140403638T449132p10T6+p20T8 1 - 49132 T^{2} + 1140403638 T^{4} - 49132 p^{10} T^{6} + p^{20} T^{8}
11C22C2C_2^2 \wr C_2 1617088T2+146890430542T4617088p10T6+p20T8 1 - 617088 T^{2} + 146890430542 T^{4} - 617088 p^{10} T^{6} + p^{20} T^{8}
17D4D_{4} (1192T+2791006T2192p5T3+p10T4)2 ( 1 - 192 T + 2791006 T^{2} - 192 p^{5} T^{3} + p^{10} T^{4} )^{2}
19C22C2C_2^2 \wr C_2 14292764T2+9023685324870T44292764p10T6+p20T8 1 - 4292764 T^{2} + 9023685324870 T^{4} - 4292764 p^{10} T^{6} + p^{20} T^{8}
23D4D_{4} (1+1212T+3450766T2+1212p5T3+p10T4)2 ( 1 + 1212 T + 3450766 T^{2} + 1212 p^{5} T^{3} + p^{10} T^{4} )^{2}
29D4D_{4} (1+1980T+13967182T2+1980p5T3+p10T4)2 ( 1 + 1980 T + 13967182 T^{2} + 1980 p^{5} T^{3} + p^{10} T^{4} )^{2}
31C22C2C_2^2 \wr C_2 172558796T2+2754768158742p2T472558796p10T6+p20T8 1 - 72558796 T^{2} + 2754768158742 p^{2} T^{4} - 72558796 p^{10} T^{6} + p^{20} T^{8}
37C22C2C_2^2 \wr C_2 1191019364T2+17466588618543606T4191019364p10T6+p20T8 1 - 191019364 T^{2} + 17466588618543606 T^{4} - 191019364 p^{10} T^{6} + p^{20} T^{8}
41C22C2C_2^2 \wr C_2 1259347072T2+41835734180463454T4259347072p10T6+p20T8 1 - 259347072 T^{2} + 41835734180463454 T^{4} - 259347072 p^{10} T^{6} + p^{20} T^{8}
43D4D_{4} (1+2416T+284123046T2+2416p5T3+p10T4)2 ( 1 + 2416 T + 284123046 T^{2} + 2416 p^{5} T^{3} + p^{10} T^{4} )^{2}
47C22C2C_2^2 \wr C_2 1407420832T2+124761975513464638T4407420832p10T6+p20T8 1 - 407420832 T^{2} + 124761975513464638 T^{4} - 407420832 p^{10} T^{6} + p^{20} T^{8}
53D4D_{4} (1+912T+836540998T2+912p5T3+p10T4)2 ( 1 + 912 T + 836540998 T^{2} + 912 p^{5} T^{3} + p^{10} T^{4} )^{2}
59C22C2C_2^2 \wr C_2 1151308144T2+442844399562129262T4151308144p10T6+p20T8 1 - 151308144 T^{2} + 442844399562129262 T^{4} - 151308144 p^{10} T^{6} + p^{20} T^{8}
61D4D_{4} (1+50496T+2324330710T2+50496p5T3+p10T4)2 ( 1 + 50496 T + 2324330710 T^{2} + 50496 p^{5} T^{3} + p^{10} T^{4} )^{2}
67C22C2C_2^2 \wr C_2 145114940T2516263867957300538T445114940p10T6+p20T8 1 - 45114940 T^{2} - 516263867957300538 T^{4} - 45114940 p^{10} T^{6} + p^{20} T^{8}
71C22C2C_2^2 \wr C_2 16784475376T2+18015098428504544062T46784475376p10T6+p20T8 1 - 6784475376 T^{2} + 18015098428504544062 T^{4} - 6784475376 p^{10} T^{6} + p^{20} T^{8}
73C22C2C_2^2 \wr C_2 1+4119671084T2+12172896815453134278T4+4119671084p10T6+p20T8 1 + 4119671084 T^{2} + 12172896815453134278 T^{4} + 4119671084 p^{10} T^{6} + p^{20} T^{8}
79D4D_{4} (1102672T+7610525470T2102672p5T3+p10T4)2 ( 1 - 102672 T + 7610525470 T^{2} - 102672 p^{5} T^{3} + p^{10} T^{4} )^{2}
83C22C2C_2^2 \wr C_2 112846419184T2+72288762441539676238T412846419184p10T6+p20T8 1 - 12846419184 T^{2} + 72288762441539676238 T^{4} - 12846419184 p^{10} T^{6} + p^{20} T^{8}
89C22C2C_2^2 \wr C_2 1+9461922432T2+70529599078905653662T4+9461922432p10T6+p20T8 1 + 9461922432 T^{2} + 70529599078905653662 T^{4} + 9461922432 p^{10} T^{6} + p^{20} T^{8}
97C22C2C_2^2 \wr C_2 123763592180T2+ 1 - 23763592180 T^{2} + 28 ⁣ ⁣9428\!\cdots\!94T423763592180p10T6+p20T8 T^{4} - 23763592180 p^{10} T^{6} + p^{20} T^{8}
show more
show less
   L(s)=p j=18(1αj,pps)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

−6.73548468769692341503113403145, −6.58480493649862435851448345709, −6.29614761626982955008685865880, −6.06465064269600518032788290438, −5.81543646847883110422149216338, −5.71133189284422302858144532982, −5.53010982580691734201868019258, −5.14791063465802708158219883836, −4.89284365950255543310569194522, −4.70802418925461832163991844027, −4.58801467785745897756071883492, −4.13507870525019502984986974555, −4.02465113623678247740999029633, −3.58121956570458921271735987118, −3.43012528962717217230170215135, −3.12873385530515772411013333359, −2.55035684222680872200842907481, −2.48837013734943451908246258424, −1.92657821483756308793116593908, −1.65930055932560466870314747301, −1.42824408489393974091797167423, −0.987217426542848906393094559167, −0.70759572161363514706061269632, −0.62804971296690877803214317371, −0.11036701592707312095807058524, 0.11036701592707312095807058524, 0.62804971296690877803214317371, 0.70759572161363514706061269632, 0.987217426542848906393094559167, 1.42824408489393974091797167423, 1.65930055932560466870314747301, 1.92657821483756308793116593908, 2.48837013734943451908246258424, 2.55035684222680872200842907481, 3.12873385530515772411013333359, 3.43012528962717217230170215135, 3.58121956570458921271735987118, 4.02465113623678247740999029633, 4.13507870525019502984986974555, 4.58801467785745897756071883492, 4.70802418925461832163991844027, 4.89284365950255543310569194522, 5.14791063465802708158219883836, 5.53010982580691734201868019258, 5.71133189284422302858144532982, 5.81543646847883110422149216338, 6.06465064269600518032788290438, 6.29614761626982955008685865880, 6.58480493649862435851448345709, 6.73548468769692341503113403145

Graph of the ZZ-function along the critical line