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Properties

Label 16-714e8-1.1-c1e8-0-1
Degree $16$
Conductor $6.754\times 10^{22}$
Sign $1$
Analytic cond. $1.11635\times 10^{6}$
Root an. cond. $2.38774$
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  + 2·4-s + 4·11-s + 8·13-s + 16-s + 4·23-s − 32·29-s − 4·31-s − 20·37-s − 16·41-s + 8·44-s + 20·47-s + 16·52-s − 4·61-s − 2·64-s − 24·67-s − 40·71-s − 20·73-s + 81-s − 16·89-s + 8·92-s + 48·103-s + 32·107-s − 32·113-s − 64·116-s + 8·121-s − 8·124-s + 127-s + ⋯
L(s)  = 1  + 4-s + 1.20·11-s + 2.21·13-s + 1/4·16-s + 0.834·23-s − 5.94·29-s − 0.718·31-s − 3.28·37-s − 2.49·41-s + 1.20·44-s + 2.91·47-s + 2.21·52-s − 0.512·61-s − 1/4·64-s − 2.93·67-s − 4.74·71-s − 2.34·73-s + 1/9·81-s − 1.69·89-s + 0.834·92-s + 4.72·103-s + 3.09·107-s − 3.01·113-s − 5.94·116-s + 8/11·121-s − 0.718·124-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(1.11635\times 10^{6}\)
Root analytic conductor: \(2.38774\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 17^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2993171431\)
\(L(\frac12)\) \(\approx\) \(0.2993171431\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} )^{2} \)
3 \( 1 - T^{4} + T^{8} \)
7 \( 1 + 2 T^{4} + p^{4} T^{8} \)
17 \( 1 - 16 T^{2} - 33 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \)
good5 \( 1 - 31 T^{4} + 336 T^{8} - 31 p^{4} T^{12} + p^{8} T^{16} \)
11 \( 1 - 4 T + 8 T^{2} - 56 T^{3} + 290 T^{4} - 1196 T^{5} + 4032 T^{6} - 13444 T^{7} + 50307 T^{8} - 13444 p T^{9} + 4032 p^{2} T^{10} - 1196 p^{3} T^{11} + 290 p^{4} T^{12} - 56 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
13 \( ( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \)
19 \( ( 1 + 22 T^{2} + 123 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( 1 - 4 T + 8 T^{2} - 120 T^{3} + 1154 T^{4} - 5948 T^{5} + 21760 T^{6} - 122004 T^{7} + 866515 T^{8} - 122004 p T^{9} + 21760 p^{2} T^{10} - 5948 p^{3} T^{11} + 1154 p^{4} T^{12} - 120 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
29 \( ( 1 + 16 T + 128 T^{2} + 832 T^{3} + 4879 T^{4} + 832 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( 1 + 4 T + 8 T^{2} - 160 T^{3} - 1423 T^{4} - 5080 T^{5} + 3864 T^{6} + 208188 T^{7} + 899200 T^{8} + 208188 p T^{9} + 3864 p^{2} T^{10} - 5080 p^{3} T^{11} - 1423 p^{4} T^{12} - 160 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 + 20 T + 200 T^{2} + 680 T^{3} - 4862 T^{4} - 73060 T^{5} - 257600 T^{6} + 1279980 T^{7} + 17933283 T^{8} + 1279980 p T^{9} - 257600 p^{2} T^{10} - 73060 p^{3} T^{11} - 4862 p^{4} T^{12} + 680 p^{5} T^{13} + 200 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \)
41 \( ( 1 + 8 T + 32 T^{2} - p^{2} T^{4} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 8 T^{2} - 894 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 - 10 T - T^{2} - 70 T^{3} + 3292 T^{4} - 70 p T^{5} - p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( 1 + 136 T^{2} + 8542 T^{4} + 589696 T^{6} + 39236371 T^{8} + 589696 p^{2} T^{10} + 8542 p^{4} T^{12} + 136 p^{6} T^{14} + p^{8} T^{16} \)
59 \( 1 + 182 T^{2} + 18081 T^{4} + 1470742 T^{6} + 98869844 T^{8} + 1470742 p^{2} T^{10} + 18081 p^{4} T^{12} + 182 p^{6} T^{14} + p^{8} T^{16} \)
61 \( 1 + 4 T + 8 T^{2} - 344 T^{3} - 4910 T^{4} - 20804 T^{5} + 15232 T^{6} + 1637244 T^{7} + 8848707 T^{8} + 1637244 p T^{9} + 15232 p^{2} T^{10} - 20804 p^{3} T^{11} - 4910 p^{4} T^{12} - 344 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
67 \( ( 1 + 12 T - 18 T^{2} + 336 T^{3} + 12107 T^{4} + 336 p T^{5} - 18 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 20 T + 200 T^{2} + 2340 T^{3} + 25262 T^{4} + 2340 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 + 20 T + 200 T^{2} - 760 T^{3} - 33806 T^{4} - 398500 T^{5} - 920000 T^{6} + 21970620 T^{7} + 342908595 T^{8} + 21970620 p T^{9} - 920000 p^{2} T^{10} - 398500 p^{3} T^{11} - 33806 p^{4} T^{12} - 760 p^{5} T^{13} + 200 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 2402 T^{4} - 33180477 T^{8} - 2402 p^{4} T^{12} + p^{8} T^{16} \)
83 \( ( 1 - 86 T^{2} + 14659 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 8 T - 98 T^{2} - 128 T^{3} + 12627 T^{4} - 128 p T^{5} - 98 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 16318 T^{4} + p^{4} T^{8} )^{2} \)
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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

−4.44047058670285849992853263295, −4.30436598043258176701666458879, −4.12292045374584916296917832734, −4.02697735879659951931484114419, −3.95816620379385989645311000468, −3.76124446054696414730355300756, −3.74893026083380541206645678171, −3.50941139629993951997580668126, −3.46622603446328770441750246292, −3.32300055699858274860647463206, −3.11970501902739375802464857248, −3.07870366935278180604352741720, −2.81777626920706703953326131336, −2.65996232669202958307926568611, −2.60669101161777246558910054028, −2.02681618521072666992803375623, −1.98337057928675835338830489804, −1.95044381871835683909138393845, −1.63109753450818900483209509434, −1.55281859388991218743095544190, −1.54755760106310625322695031947, −1.48373649611569649848319343858, −0.996614942499960636697752162216, −0.54490323116024875562448223345, −0.07244616968423554514348862813, 0.07244616968423554514348862813, 0.54490323116024875562448223345, 0.996614942499960636697752162216, 1.48373649611569649848319343858, 1.54755760106310625322695031947, 1.55281859388991218743095544190, 1.63109753450818900483209509434, 1.95044381871835683909138393845, 1.98337057928675835338830489804, 2.02681618521072666992803375623, 2.60669101161777246558910054028, 2.65996232669202958307926568611, 2.81777626920706703953326131336, 3.07870366935278180604352741720, 3.11970501902739375802464857248, 3.32300055699858274860647463206, 3.46622603446328770441750246292, 3.50941139629993951997580668126, 3.74893026083380541206645678171, 3.76124446054696414730355300756, 3.95816620379385989645311000468, 4.02697735879659951931484114419, 4.12292045374584916296917832734, 4.30436598043258176701666458879, 4.44047058670285849992853263295

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

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