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Properties

Label 16-1638e8-1.1-c1e8-0-7
Degree $16$
Conductor $5.182\times 10^{25}$
Sign $1$
Analytic cond. $8.56501\times 10^{8}$
Root an. cond. $3.61655$
Motivic weight $1$
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  + 4·5-s + 16·13-s − 2·16-s − 4·17-s − 8·19-s + 8·25-s − 12·29-s − 8·31-s − 4·37-s − 12·41-s + 16·49-s − 40·53-s + 8·59-s + 64·65-s + 32·67-s + 12·71-s + 20·73-s + 24·79-s − 8·80-s − 44·83-s − 16·85-s − 16·89-s − 32·95-s − 8·97-s + 16·101-s + 12·103-s − 8·107-s + ⋯
L(s)  = 1  + 1.78·5-s + 4.43·13-s − 1/2·16-s − 0.970·17-s − 1.83·19-s + 8/5·25-s − 2.22·29-s − 1.43·31-s − 0.657·37-s − 1.87·41-s + 16/7·49-s − 5.49·53-s + 1.04·59-s + 7.93·65-s + 3.90·67-s + 1.42·71-s + 2.34·73-s + 2.70·79-s − 0.894·80-s − 4.82·83-s − 1.73·85-s − 1.69·89-s − 3.28·95-s − 0.812·97-s + 1.59·101-s + 1.18·103-s − 0.773·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(8.56501\times 10^{8}\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.713618063\)
\(L(\frac12)\) \(\approx\) \(2.713618063\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{4} )^{2} \)
3 \( 1 \)
7 \( 1 - 16 T^{2} + 130 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \)
13 \( 1 - 16 T + 114 T^{2} - 512 T^{3} + 1890 T^{4} - 512 p T^{5} + 114 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
good5 \( 1 - 4 T + 8 T^{2} - 16 T^{3} + 41 T^{4} - 128 T^{5} + 312 T^{6} - 828 T^{7} + 2176 T^{8} - 828 p T^{9} + 312 p^{2} T^{10} - 128 p^{3} T^{11} + 41 p^{4} T^{12} - 16 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 + 48 T^{3} + 57 T^{4} - 768 T^{5} + 1152 T^{6} - 1872 T^{7} - 34240 T^{8} - 1872 p T^{9} + 1152 p^{2} T^{10} - 768 p^{3} T^{11} + 57 p^{4} T^{12} + 48 p^{5} T^{13} + p^{8} T^{16} \)
17 \( ( 1 + 2 T + 45 T^{2} + 134 T^{3} + 944 T^{4} + 134 p T^{5} + 45 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( 1 + 8 T + 32 T^{2} + 176 T^{3} + 51 p T^{4} + 3200 T^{5} + 10080 T^{6} + 36696 T^{7} + 117184 T^{8} + 36696 p T^{9} + 10080 p^{2} T^{10} + 3200 p^{3} T^{11} + 51 p^{5} T^{12} + 176 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 38 T^{2} + 1333 T^{4} - 32994 T^{6} + 926696 T^{8} - 32994 p^{2} T^{10} + 1333 p^{4} T^{12} - 38 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 + 6 T + 71 T^{2} + 298 T^{3} + 2534 T^{4} + 298 p T^{5} + 71 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( 1 + 8 T + 32 T^{2} + 216 T^{3} + 2372 T^{4} + 14392 T^{5} + 62560 T^{6} + 474792 T^{7} + 3604870 T^{8} + 474792 p T^{9} + 62560 p^{2} T^{10} + 14392 p^{3} T^{11} + 2372 p^{4} T^{12} + 216 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 + 4 T + 8 T^{2} + 112 T^{3} + 3465 T^{4} + 11872 T^{5} + 26040 T^{6} + 394524 T^{7} + 5896384 T^{8} + 394524 p T^{9} + 26040 p^{2} T^{10} + 11872 p^{3} T^{11} + 3465 p^{4} T^{12} + 112 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 + 12 T + 72 T^{2} + 436 T^{3} + 2176 T^{4} + 11692 T^{5} + 78680 T^{6} + 488660 T^{7} + 3032766 T^{8} + 488660 p T^{9} + 78680 p^{2} T^{10} + 11692 p^{3} T^{11} + 2176 p^{4} T^{12} + 436 p^{5} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 - 6 p T^{2} + 31169 T^{4} - 2332954 T^{6} + 119560116 T^{8} - 2332954 p^{2} T^{10} + 31169 p^{4} T^{12} - 6 p^{7} T^{14} + p^{8} T^{16} \)
47 \( ( 1 + 3682 T^{4} + p^{4} T^{8} )^{2} \)
53 \( ( 1 + 20 T + 278 T^{2} + 2892 T^{3} + 23978 T^{4} + 2892 p T^{5} + 278 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 - 8 T + 32 T^{2} + 248 T^{3} + 2076 T^{4} + 7864 T^{5} - 98592 T^{6} + 1411448 T^{7} + 8207270 T^{8} + 1411448 p T^{9} - 98592 p^{2} T^{10} + 7864 p^{3} T^{11} + 2076 p^{4} T^{12} + 248 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 + 14 T^{2} + 7685 T^{4} + 105546 T^{6} + 30490872 T^{8} + 105546 p^{2} T^{10} + 7685 p^{4} T^{12} + 14 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 32 T + 512 T^{2} - 5600 T^{3} + 50684 T^{4} - 415584 T^{5} + 3028480 T^{6} - 18496928 T^{7} + 122079910 T^{8} - 18496928 p T^{9} + 3028480 p^{2} T^{10} - 415584 p^{3} T^{11} + 50684 p^{4} T^{12} - 5600 p^{5} T^{13} + 512 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 12 T + 72 T^{2} - 908 T^{3} + 2064 T^{4} + 51444 T^{5} - 353704 T^{6} + 5033588 T^{7} - 69761186 T^{8} + 5033588 p T^{9} - 353704 p^{2} T^{10} + 51444 p^{3} T^{11} + 2064 p^{4} T^{12} - 908 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 - 20 T + 200 T^{2} - 576 T^{3} - 9079 T^{4} + 107216 T^{5} - 162632 T^{6} - 7946652 T^{7} + 106585744 T^{8} - 7946652 p T^{9} - 162632 p^{2} T^{10} + 107216 p^{3} T^{11} - 9079 p^{4} T^{12} - 576 p^{5} T^{13} + 200 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \)
79 \( ( 1 - 12 T + 246 T^{2} - 1836 T^{3} + 318 p T^{4} - 1836 p T^{5} + 246 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 + 44 T + 968 T^{2} + 16060 T^{3} + 235728 T^{4} + 3037100 T^{5} + 34409496 T^{6} + 357330556 T^{7} + 3416693630 T^{8} + 357330556 p T^{9} + 34409496 p^{2} T^{10} + 3037100 p^{3} T^{11} + 235728 p^{4} T^{12} + 16060 p^{5} T^{13} + 968 p^{6} T^{14} + 44 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 + 16 T + 128 T^{2} + 1328 T^{3} + 23068 T^{4} + 260752 T^{5} + 2101120 T^{6} + 20928816 T^{7} + 206877318 T^{8} + 20928816 p T^{9} + 2101120 p^{2} T^{10} + 260752 p^{3} T^{11} + 23068 p^{4} T^{12} + 1328 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 + 8 T + 32 T^{2} + 1032 T^{3} - 988 T^{4} - 101096 T^{5} - 244640 T^{6} - 6736104 T^{7} - 156517562 T^{8} - 6736104 p T^{9} - 244640 p^{2} T^{10} - 101096 p^{3} T^{11} - 988 p^{4} T^{12} + 1032 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
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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

−3.98255338742910584107044327869, −3.72326657862004871183007870949, −3.69299005873703833046718096114, −3.53820394966285664404923181467, −3.50722537720726918058196148133, −3.35748159278668978846734250887, −3.22855152109419200378245752218, −3.22783872609124218754817339725, −3.09408804758013918751245676679, −2.79473337700238262918104675268, −2.49259523678054783738941736073, −2.31986786954912968597599522708, −2.28268620356781601687882686831, −2.24358506909379042911250269014, −2.10996620570150460876557073760, −2.00903881593079046388107339821, −1.81787317726074546486800321538, −1.51715049398176134631905184990, −1.49134742088606634988512806481, −1.34154451992563246736847127679, −1.27656675500465587345892387259, −1.10804022053835977901716522133, −0.56107084863285174952276971209, −0.55158528342482315144556995103, −0.13192398503991023497189458915, 0.13192398503991023497189458915, 0.55158528342482315144556995103, 0.56107084863285174952276971209, 1.10804022053835977901716522133, 1.27656675500465587345892387259, 1.34154451992563246736847127679, 1.49134742088606634988512806481, 1.51715049398176134631905184990, 1.81787317726074546486800321538, 2.00903881593079046388107339821, 2.10996620570150460876557073760, 2.24358506909379042911250269014, 2.28268620356781601687882686831, 2.31986786954912968597599522708, 2.49259523678054783738941736073, 2.79473337700238262918104675268, 3.09408804758013918751245676679, 3.22783872609124218754817339725, 3.22855152109419200378245752218, 3.35748159278668978846734250887, 3.50722537720726918058196148133, 3.53820394966285664404923181467, 3.69299005873703833046718096114, 3.72326657862004871183007870949, 3.98255338742910584107044327869

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

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