Properties

Label 16-300e8-1.1-c2e8-0-5
Degree $16$
Conductor $6.561\times 10^{19}$
Sign $1$
Analytic cond. $1.99365\times 10^{7}$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·4-s − 12·9-s − 8·13-s + 24·16-s + 24·18-s + 16·26-s − 32·29-s − 40·32-s + 24·36-s + 176·37-s − 16·41-s + 204·49-s + 16·52-s + 304·53-s + 64·58-s + 136·61-s + 8·64-s − 240·73-s − 352·74-s + 90·81-s + 32·82-s + 128·89-s − 216·97-s − 408·98-s + 112·101-s − 608·106-s + ⋯
L(s)  = 1  − 2-s − 1/2·4-s − 4/3·9-s − 0.615·13-s + 3/2·16-s + 4/3·18-s + 8/13·26-s − 1.10·29-s − 5/4·32-s + 2/3·36-s + 4.75·37-s − 0.390·41-s + 4.16·49-s + 4/13·52-s + 5.73·53-s + 1.10·58-s + 2.22·61-s + 1/8·64-s − 3.28·73-s − 4.75·74-s + 10/9·81-s + 0.390·82-s + 1.43·89-s − 2.22·97-s − 4.16·98-s + 1.10·101-s − 5.73·106-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(1.99365\times 10^{7}\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.655371595\)
\(L(\frac12)\) \(\approx\) \(1.655371595\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T + 3 p T^{2} + p^{4} T^{3} + 5 p^{2} T^{4} + p^{6} T^{5} + 3 p^{5} T^{6} + p^{7} T^{7} + p^{8} T^{8} \)
3 \( ( 1 + p T^{2} )^{4} \)
5 \( 1 \)
good7 \( 1 - 204 T^{2} + 18426 T^{4} - 1076112 T^{6} + 53449475 T^{8} - 1076112 p^{4} T^{10} + 18426 p^{8} T^{12} - 204 p^{12} T^{14} + p^{16} T^{16} \)
11 \( 1 - 24 p T^{2} + 33852 T^{4} - 3268920 T^{6} + 337315526 T^{8} - 3268920 p^{4} T^{10} + 33852 p^{8} T^{12} - 24 p^{13} T^{14} + p^{16} T^{16} \)
13 \( ( 1 + 4 T + 258 T^{2} - 1888 T^{3} + 28523 T^{4} - 1888 p^{2} T^{5} + 258 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
17 \( ( 1 + 324 T^{2} - 1920 T^{3} + 152630 T^{4} - 1920 p^{2} T^{5} + 324 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
19 \( 1 - 1756 T^{2} + 1510026 T^{4} - 44288912 p T^{6} + 346336758035 T^{8} - 44288912 p^{5} T^{10} + 1510026 p^{8} T^{12} - 1756 p^{12} T^{14} + p^{16} T^{16} \)
23 \( 1 - 2248 T^{2} + 2472828 T^{4} - 1852675064 T^{6} + 1086691824134 T^{8} - 1852675064 p^{4} T^{10} + 2472828 p^{8} T^{12} - 2248 p^{12} T^{14} + p^{16} T^{16} \)
29 \( ( 1 + 16 T + 1764 T^{2} + 10544 T^{3} + 1646102 T^{4} + 10544 p^{2} T^{5} + 1764 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
31 \( 1 - 2028 T^{2} + 2805402 T^{4} - 2884712400 T^{6} + 3076391251619 T^{8} - 2884712400 p^{4} T^{10} + 2805402 p^{8} T^{12} - 2028 p^{12} T^{14} + p^{16} T^{16} \)
37 \( ( 1 - 88 T + 4764 T^{2} - 139688 T^{3} + 4747046 T^{4} - 139688 p^{2} T^{5} + 4764 p^{4} T^{6} - 88 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( ( 1 + 8 T + 1756 T^{2} + 78008 T^{3} + 3756598 T^{4} + 78008 p^{2} T^{5} + 1756 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
43 \( 1 - 7900 T^{2} + 33132810 T^{4} - 96255124016 T^{6} + 206878197918227 T^{8} - 96255124016 p^{4} T^{10} + 33132810 p^{8} T^{12} - 7900 p^{12} T^{14} + p^{16} T^{16} \)
47 \( 1 - 5656 T^{2} + 20883420 T^{4} - 53920616360 T^{6} + 122334534618566 T^{8} - 53920616360 p^{4} T^{10} + 20883420 p^{8} T^{12} - 5656 p^{12} T^{14} + p^{16} T^{16} \)
53 \( ( 1 - 152 T + 15804 T^{2} - 1145512 T^{3} + 68059286 T^{4} - 1145512 p^{2} T^{5} + 15804 p^{4} T^{6} - 152 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
59 \( 1 - 21656 T^{2} + 220430236 T^{4} - 1377870886568 T^{6} + 5784129582348550 T^{8} - 1377870886568 p^{4} T^{10} + 220430236 p^{8} T^{12} - 21656 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 - 68 T + 6786 T^{2} - 434848 T^{3} + 32554859 T^{4} - 434848 p^{2} T^{5} + 6786 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( 1 - 14364 T^{2} + 117868746 T^{4} - 700523495472 T^{6} + 3348544470504275 T^{8} - 700523495472 p^{4} T^{10} + 117868746 p^{8} T^{12} - 14364 p^{12} T^{14} + p^{16} T^{16} \)
71 \( 1 - 31512 T^{2} + 469696188 T^{4} - 4285342561320 T^{6} + 26126405445829766 T^{8} - 4285342561320 p^{4} T^{10} + 469696188 p^{8} T^{12} - 31512 p^{12} T^{14} + p^{16} T^{16} \)
73 \( ( 1 + 120 T + 12732 T^{2} + 423240 T^{3} + 35299142 T^{4} + 423240 p^{2} T^{5} + 12732 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( 1 - 35400 T^{2} + 591951516 T^{4} - 6232687244280 T^{6} + 45861936410406470 T^{8} - 6232687244280 p^{4} T^{10} + 591951516 p^{8} T^{12} - 35400 p^{12} T^{14} + p^{16} T^{16} \)
83 \( 1 - 6424 T^{2} + 64588572 T^{4} - 770632625960 T^{6} + 4303554072892550 T^{8} - 770632625960 p^{4} T^{10} + 64588572 p^{8} T^{12} - 6424 p^{12} T^{14} + p^{16} T^{16} \)
89 \( ( 1 - 64 T + 28356 T^{2} - 1283264 T^{3} + 322223942 T^{4} - 1283264 p^{2} T^{5} + 28356 p^{4} T^{6} - 64 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
97 \( ( 1 + 108 T + 26650 T^{2} + 2076624 T^{3} + 340059171 T^{4} + 2076624 p^{2} T^{5} + 26650 p^{4} T^{6} + 108 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
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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

−5.25625476732129540933538406061, −4.79843528605876422811484743095, −4.53261217976029711841356602374, −4.45923241067128923285733526428, −4.45353987570142024503241733034, −4.37356569518178441379718868532, −3.90543665000124501326175068350, −3.83307896495019608215957845310, −3.80110532091597969833679552851, −3.62752075446351502791654068577, −3.61796906999894973479839425140, −3.29272807128251889904371969013, −2.68889461451941087253264360499, −2.67281635829614390259334939299, −2.66902590430904938673351027541, −2.60029944476819899145037096747, −2.50110731834166046082714873565, −2.25619462362142325213700437744, −1.86604810252440870234737898652, −1.55115186131344348395923547764, −1.14132465030501480292308923314, −1.02052608948419132526273470163, −0.72229386621204027289868711437, −0.45268822310522814203493158010, −0.34246291222299185279104796929, 0.34246291222299185279104796929, 0.45268822310522814203493158010, 0.72229386621204027289868711437, 1.02052608948419132526273470163, 1.14132465030501480292308923314, 1.55115186131344348395923547764, 1.86604810252440870234737898652, 2.25619462362142325213700437744, 2.50110731834166046082714873565, 2.60029944476819899145037096747, 2.66902590430904938673351027541, 2.67281635829614390259334939299, 2.68889461451941087253264360499, 3.29272807128251889904371969013, 3.61796906999894973479839425140, 3.62752075446351502791654068577, 3.80110532091597969833679552851, 3.83307896495019608215957845310, 3.90543665000124501326175068350, 4.37356569518178441379718868532, 4.45353987570142024503241733034, 4.45923241067128923285733526428, 4.53261217976029711841356602374, 4.79843528605876422811484743095, 5.25625476732129540933538406061

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

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