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Properties

Label 16-300e8-1.1-c6e8-0-0
Degree $16$
Conductor $6.561\times 10^{19}$
Sign $1$
Analytic cond. $5.14765\times 10^{14}$
Root an. cond. $8.30760$
Motivic weight $6$
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  − 912·11-s + 5.48e4·31-s − 2.48e5·41-s − 6.71e5·61-s − 1.16e6·71-s − 1.18e5·81-s − 1.00e7·101-s − 1.09e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 0.685·11-s + 1.84·31-s − 3.60·41-s − 2.95·61-s − 3.26·71-s − 2/9·81-s − 9.73·101-s − 6.17·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(5.14765\times 10^{14}\)
Root analytic conductor: \(8.30760\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [3]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.07136247733\)
\(L(\frac12)\) \(\approx\) \(0.07136247733\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + p^{10} T^{4} )^{2} \)
5 \( 1 \)
good7 \( 1 - 6168503246 p T^{4} + \)\(84\!\cdots\!23\)\( T^{8} - 6168503246 p^{25} T^{12} + p^{48} T^{16} \)
11 \( ( 1 + 228 T + 2866718 T^{2} + 228 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
13 \( 1 + 37946640103774 T^{4} + \)\(68\!\cdots\!91\)\( T^{8} + 37946640103774 p^{24} T^{12} + p^{48} T^{16} \)
17 \( 1 - 568503779803772 T^{4} + \)\(73\!\cdots\!38\)\( T^{8} - 568503779803772 p^{24} T^{12} + p^{48} T^{16} \)
19 \( ( 1 - 180288674 T^{2} + 12543647001561891 T^{4} - 180288674 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
23 \( 1 + 15021586583776516 T^{4} - \)\(22\!\cdots\!54\)\( T^{8} + 15021586583776516 p^{24} T^{12} + p^{48} T^{16} \)
29 \( ( 1 - 1535709676 T^{2} + 1244801613539574726 T^{4} - 1535709676 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
31 \( ( 1 - 13718 T + 1815848643 T^{2} - 13718 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
37 \( 1 - 8133251316795870524 T^{4} + \)\(56\!\cdots\!66\)\( T^{8} - 8133251316795870524 p^{24} T^{12} + p^{48} T^{16} \)
41 \( ( 1 + 62160 T + 372669482 T^{2} + 62160 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
43 \( 1 - 59679583497807293954 T^{4} + \)\(24\!\cdots\!31\)\( T^{8} - 59679583497807293954 p^{24} T^{12} + p^{48} T^{16} \)
47 \( 1 + \)\(42\!\cdots\!68\)\( T^{4} + \)\(71\!\cdots\!18\)\( T^{8} + \)\(42\!\cdots\!68\)\( p^{24} T^{12} + p^{48} T^{16} \)
53 \( 1 - \)\(41\!\cdots\!52\)\( T^{4} + \)\(48\!\cdots\!38\)\( T^{8} - \)\(41\!\cdots\!52\)\( p^{24} T^{12} + p^{48} T^{16} \)
59 \( ( 1 - 51610470556 T^{2} + \)\(26\!\cdots\!46\)\( T^{4} - 51610470556 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
61 \( ( 1 + 167786 T + 54474537771 T^{2} + 167786 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
67 \( 1 + \)\(17\!\cdots\!38\)\( T^{4} + \)\(16\!\cdots\!03\)\( T^{8} + \)\(17\!\cdots\!38\)\( p^{24} T^{12} + p^{48} T^{16} \)
71 \( ( 1 + 291840 T + 273190668842 T^{2} + 291840 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
73 \( 1 + \)\(10\!\cdots\!08\)\( T^{4} - \)\(36\!\cdots\!02\)\( T^{8} + \)\(10\!\cdots\!08\)\( p^{24} T^{12} + p^{48} T^{16} \)
79 \( ( 1 - 812607963452 T^{2} + \)\(27\!\cdots\!58\)\( T^{4} - 812607963452 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
83 \( 1 + \)\(17\!\cdots\!56\)\( T^{4} - \)\(61\!\cdots\!74\)\( T^{8} + \)\(17\!\cdots\!56\)\( p^{24} T^{12} + p^{48} T^{16} \)
89 \( ( 1 - 746931903844 T^{2} + \)\(25\!\cdots\!26\)\( T^{4} - 746931903844 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
97 \( 1 + \)\(13\!\cdots\!86\)\( T^{4} - \)\(16\!\cdots\!89\)\( T^{8} + \)\(13\!\cdots\!86\)\( p^{24} T^{12} + p^{48} T^{16} \)
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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

−4.13772133741885449798549450582, −4.06649326739019397287315271205, −3.93747154392343909914275209192, −3.64788451343363926699103671833, −3.55250205156993698669821072306, −3.28953217407988758412650641494, −3.05805058602160594362652153292, −3.05707319445131726638250515189, −2.78633322041436729159068619863, −2.74927711138871889500779060009, −2.71932036844286314991625374391, −2.55524746442731388850293909063, −2.18539445806422874399571473499, −2.15637989779576914201147552382, −1.80888509107863774243309697798, −1.53757160636637043888715249686, −1.46193997443259322436426937824, −1.38743836914529135222927998598, −1.36334538015177789673209438726, −1.04470281191089952416827342225, −1.03653726747423936558061563987, −0.53005135457318538197047701567, −0.28481697366766151954102863729, −0.18737903895014587695585862141, −0.04010105732999288879812191853, 0.04010105732999288879812191853, 0.18737903895014587695585862141, 0.28481697366766151954102863729, 0.53005135457318538197047701567, 1.03653726747423936558061563987, 1.04470281191089952416827342225, 1.36334538015177789673209438726, 1.38743836914529135222927998598, 1.46193997443259322436426937824, 1.53757160636637043888715249686, 1.80888509107863774243309697798, 2.15637989779576914201147552382, 2.18539445806422874399571473499, 2.55524746442731388850293909063, 2.71932036844286314991625374391, 2.74927711138871889500779060009, 2.78633322041436729159068619863, 3.05707319445131726638250515189, 3.05805058602160594362652153292, 3.28953217407988758412650641494, 3.55250205156993698669821072306, 3.64788451343363926699103671833, 3.93747154392343909914275209192, 4.06649326739019397287315271205, 4.13772133741885449798549450582

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

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