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Properties

Label 16-162e8-1.1-c10e8-0-1
Degree $16$
Conductor $4.744\times 10^{17}$
Sign $1$
Analytic cond. $1.25969\times 10^{16}$
Root an. cond. $10.1453$
Motivic weight $10$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.02e3·4-s − 1.15e4·7-s + 1.35e6·13-s + 2.62e5·16-s − 1.69e6·19-s + 3.72e6·25-s − 1.17e7·28-s + 4.31e7·31-s − 6.32e7·37-s − 5.43e8·43-s + 6.75e8·49-s + 1.38e9·52-s − 1.62e9·61-s − 2.68e8·64-s + 6.66e9·67-s − 2.19e10·73-s − 1.73e9·76-s + 1.63e9·79-s − 1.55e10·91-s − 2.10e10·97-s + 3.81e9·100-s + 2.79e10·103-s − 9.03e10·109-s − 3.01e9·112-s − 6.62e10·121-s + 4.41e10·124-s + 127-s + ⋯
L(s)  = 1  + 4-s − 0.685·7-s + 3.63·13-s + 1/4·16-s − 0.685·19-s + 0.381·25-s − 0.685·28-s + 1.50·31-s − 0.912·37-s − 3.69·43-s + 2.39·49-s + 3.63·52-s − 1.92·61-s − 1/4·64-s + 4.93·67-s − 10.6·73-s − 0.685·76-s + 0.531·79-s − 2.49·91-s − 2.44·97-s + 0.381·100-s + 2.41·103-s − 5.87·109-s − 0.171·112-s − 2.55·121-s + 1.50·124-s + 0.470·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+5)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{32}\)
Sign: $1$
Analytic conductor: \(1.25969\times 10^{16}\)
Root analytic conductor: \(10.1453\)
Motivic weight: \(10\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{32} ,\ ( \ : [5]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(4.605334585\)
\(L(\frac12)\) \(\approx\) \(4.605334585\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p^{9} T^{2} + p^{18} T^{4} )^{2} \)
3 \( 1 \)
good5 \( 1 - 745708 p T^{2} - 3283779575942 p^{2} T^{4} + 2825886218312883152 p^{3} T^{6} - \)\(19\!\cdots\!89\)\( p^{4} T^{8} + 2825886218312883152 p^{23} T^{10} - 3283779575942 p^{42} T^{12} - 745708 p^{61} T^{14} + p^{80} T^{16} \)
7 \( ( 1 + 5758 T - 287850575 T^{2} - 200662482446 p T^{3} + 283401821334916 p^{2} T^{4} - 200662482446 p^{11} T^{5} - 287850575 p^{20} T^{6} + 5758 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
11 \( 1 + 66243867364 T^{2} + \)\(17\!\cdots\!70\)\( p T^{4} + \)\(72\!\cdots\!36\)\( T^{6} + \)\(25\!\cdots\!79\)\( T^{8} + \)\(72\!\cdots\!36\)\( p^{20} T^{10} + \)\(17\!\cdots\!70\)\( p^{41} T^{12} + 66243867364 p^{60} T^{14} + p^{80} T^{16} \)
13 \( ( 1 - 675530 T + 80350935817 T^{2} - 67737326051409050 T^{3} + \)\(59\!\cdots\!88\)\( T^{4} - 67737326051409050 p^{10} T^{5} + 80350935817 p^{20} T^{6} - 675530 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
17 \( ( 1 - 5641479203332 T^{2} + \)\(16\!\cdots\!58\)\( T^{4} - 5641479203332 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
19 \( ( 1 + 424634 T - 1693632418149 T^{2} + 424634 p^{10} T^{3} + p^{20} T^{4} )^{4} \)
23 \( 1 + 73666728474820 T^{2} + \)\(23\!\cdots\!38\)\( T^{4} - \)\(28\!\cdots\!00\)\( T^{6} - \)\(30\!\cdots\!57\)\( T^{8} - \)\(28\!\cdots\!00\)\( p^{20} T^{10} + \)\(23\!\cdots\!38\)\( p^{40} T^{12} + 73666728474820 p^{60} T^{14} + p^{80} T^{16} \)
29 \( 1 + 1414848358082980 T^{2} + \)\(11\!\cdots\!58\)\( T^{4} + \)\(70\!\cdots\!00\)\( T^{6} + \)\(34\!\cdots\!63\)\( T^{8} + \)\(70\!\cdots\!00\)\( p^{20} T^{10} + \)\(11\!\cdots\!58\)\( p^{40} T^{12} + 1414848358082980 p^{60} T^{14} + p^{80} T^{16} \)
31 \( ( 1 - 21565916 T - 594713918938550 T^{2} + \)\(12\!\cdots\!36\)\( T^{3} + \)\(70\!\cdots\!79\)\( T^{4} + \)\(12\!\cdots\!36\)\( p^{10} T^{5} - 594713918938550 p^{20} T^{6} - 21565916 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
37 \( ( 1 + 15813050 T - 187717534860117 T^{2} + 15813050 p^{10} T^{3} + p^{20} T^{4} )^{4} \)
41 \( 1 + 34706191560202564 T^{2} + \)\(61\!\cdots\!70\)\( T^{4} + \)\(80\!\cdots\!36\)\( T^{6} + \)\(10\!\cdots\!79\)\( T^{8} + \)\(80\!\cdots\!36\)\( p^{20} T^{10} + \)\(61\!\cdots\!70\)\( p^{40} T^{12} + 34706191560202564 p^{60} T^{14} + p^{80} T^{16} \)
43 \( ( 1 + 271924756 T + 12531166778859514 T^{2} + \)\(49\!\cdots\!44\)\( T^{3} + \)\(17\!\cdots\!19\)\( T^{4} + \)\(49\!\cdots\!44\)\( p^{10} T^{5} + 12531166778859514 p^{20} T^{6} + 271924756 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
47 \( 1 + 3128280263826436 T^{2} - \)\(33\!\cdots\!30\)\( T^{4} - \)\(66\!\cdots\!36\)\( T^{6} + \)\(39\!\cdots\!79\)\( T^{8} - \)\(66\!\cdots\!36\)\( p^{20} T^{10} - \)\(33\!\cdots\!30\)\( p^{40} T^{12} + 3128280263826436 p^{60} T^{14} + p^{80} T^{16} \)
53 \( ( 1 + 45137594882174108 T^{2} + \)\(34\!\cdots\!58\)\( T^{4} + 45137594882174108 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
59 \( 1 - 656024392749515420 T^{2} - \)\(52\!\cdots\!02\)\( T^{4} + \)\(26\!\cdots\!00\)\( T^{6} + \)\(52\!\cdots\!03\)\( T^{8} + \)\(26\!\cdots\!00\)\( p^{20} T^{10} - \)\(52\!\cdots\!02\)\( p^{40} T^{12} - 656024392749515420 p^{60} T^{14} + p^{80} T^{16} \)
61 \( ( 1 + 812641750 T - 386410985135902487 T^{2} - \)\(30\!\cdots\!50\)\( T^{3} + \)\(11\!\cdots\!68\)\( T^{4} - \)\(30\!\cdots\!50\)\( p^{10} T^{5} - 386410985135902487 p^{20} T^{6} + 812641750 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
67 \( ( 1 - 3332679578 T + 5300030719262714665 T^{2} - \)\(72\!\cdots\!38\)\( T^{3} + \)\(10\!\cdots\!44\)\( T^{4} - \)\(72\!\cdots\!38\)\( p^{10} T^{5} + 5300030719262714665 p^{20} T^{6} - 3332679578 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
71 \( ( 1 - 9373061922920524228 T^{2} + \)\(39\!\cdots\!98\)\( T^{4} - 9373061922920524228 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
73 \( ( 1 + 5494029458 T + 15913367435529618099 T^{2} + 5494029458 p^{10} T^{3} + p^{20} T^{4} )^{4} \)
79 \( ( 1 - 817438898 T - 7788571657693269599 T^{2} + \)\(85\!\cdots\!02\)\( T^{3} - \)\(24\!\cdots\!96\)\( T^{4} + \)\(85\!\cdots\!02\)\( p^{10} T^{5} - 7788571657693269599 p^{20} T^{6} - 817438898 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
83 \( 1 + 56210550258075226660 T^{2} + \)\(18\!\cdots\!58\)\( T^{4} + \)\(44\!\cdots\!00\)\( T^{6} + \)\(79\!\cdots\!63\)\( T^{8} + \)\(44\!\cdots\!00\)\( p^{20} T^{10} + \)\(18\!\cdots\!58\)\( p^{40} T^{12} + 56210550258075226660 p^{60} T^{14} + p^{80} T^{16} \)
89 \( ( 1 - 86028361741737493060 T^{2} + \)\(37\!\cdots\!42\)\( T^{4} - 86028361741737493060 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
97 \( ( 1 + 10500175870 T - 64063012908380015423 T^{2} + \)\(28\!\cdots\!50\)\( T^{3} + \)\(16\!\cdots\!28\)\( T^{4} + \)\(28\!\cdots\!50\)\( p^{10} T^{5} - 64063012908380015423 p^{20} T^{6} + 10500175870 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.96824776282318844843445710801, −3.94518900122657578356147093164, −3.86207228017441322304601353440, −3.62395606743826287971011296580, −3.17334590059327419776504882397, −3.00653169456149858210395366129, −2.98554497439789753405087408223, −2.95535383916824730425028877725, −2.90596393921471223174010131845, −2.82170898614215419659412932256, −2.49212907675864319115827027738, −2.18944939459521537125367451088, −1.98402677143405750494908197770, −1.87681424262437415147484273093, −1.66590652171480221587755978896, −1.52510964079339533466172593888, −1.47518333014975739178993620289, −1.38109230636659843832281167635, −1.16568078541162237604063699544, −1.02100103938585246679689577122, −0.861483162014275715394934124921, −0.45979938185740019685422461925, −0.43754886420219931191454412738, −0.35576882851998784675561591445, −0.089945241916569744807339664440, 0.089945241916569744807339664440, 0.35576882851998784675561591445, 0.43754886420219931191454412738, 0.45979938185740019685422461925, 0.861483162014275715394934124921, 1.02100103938585246679689577122, 1.16568078541162237604063699544, 1.38109230636659843832281167635, 1.47518333014975739178993620289, 1.52510964079339533466172593888, 1.66590652171480221587755978896, 1.87681424262437415147484273093, 1.98402677143405750494908197770, 2.18944939459521537125367451088, 2.49212907675864319115827027738, 2.82170898614215419659412932256, 2.90596393921471223174010131845, 2.95535383916824730425028877725, 2.98554497439789753405087408223, 3.00653169456149858210395366129, 3.17334590059327419776504882397, 3.62395606743826287971011296580, 3.86207228017441322304601353440, 3.94518900122657578356147093164, 3.96824776282318844843445710801

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.