↑

Properties

Label 16-3150e8-1.1-c1e8-0-5
Degree $16$
Conductor $9.694\times 10^{27}$
Sign $1$
Analytic cond. $1.60214\times 10^{11}$
Root an. cond. $5.01526$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8·13-s − 2·16-s − 16·23-s + 16·37-s + 8·43-s + 8·47-s − 32·53-s + 8·59-s − 32·61-s − 16·67-s + 8·83-s − 16·89-s − 16·97-s + 8·103-s + 32·107-s − 32·113-s + 64·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + ⋯
L(s)  = 1  − 2.21·13-s − 1/2·16-s − 3.33·23-s + 2.63·37-s + 1.21·43-s + 1.16·47-s − 4.39·53-s + 1.04·59-s − 4.09·61-s − 1.95·67-s + 0.878·83-s − 1.69·89-s − 1.62·97-s + 0.788·103-s + 3.09·107-s − 3.01·113-s + 5.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.46·169-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.60214\times 10^{11}\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4833964350\)
\(L(\frac12)\) \(\approx\) \(0.4833964350\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( ( 1 + T^{4} )^{2} \)
good11 \( ( 1 - 32 T^{2} + 466 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( 1 + 8 T + 32 T^{2} + 128 T^{3} + 574 T^{4} + 2360 T^{5} + 8704 T^{6} + 35112 T^{7} + 139491 T^{8} + 35112 p T^{9} + 8704 p^{2} T^{10} + 2360 p^{3} T^{11} + 574 p^{4} T^{12} + 128 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 - 96 T^{3} + 158 T^{4} + 2400 T^{5} + 4608 T^{6} + 8544 T^{7} - 207741 T^{8} + 8544 p T^{9} + 4608 p^{2} T^{10} + 2400 p^{3} T^{11} + 158 p^{4} T^{12} - 96 p^{5} T^{13} + p^{8} T^{16} \)
19 \( ( 1 - 48 T^{2} + 1202 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( 1 + 16 T + 128 T^{2} + 800 T^{3} + 4670 T^{4} + 26576 T^{5} + 147456 T^{6} + 35120 p T^{7} + 7827 p^{2} T^{8} + 35120 p^{2} T^{9} + 147456 p^{2} T^{10} + 26576 p^{3} T^{11} + 4670 p^{4} T^{12} + 800 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
29 \( ( 1 + 2 p T^{2} - 96 T^{3} + 2019 T^{4} - 96 p T^{5} + 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 42 T^{2} - 192 T^{3} + 1139 T^{4} - 192 p T^{5} + 42 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( 1 - 16 T + 128 T^{2} - 784 T^{3} + 4324 T^{4} - 27760 T^{5} + 198016 T^{6} - 1322736 T^{7} + 8422566 T^{8} - 1322736 p T^{9} + 198016 p^{2} T^{10} - 27760 p^{3} T^{11} + 4324 p^{4} T^{12} - 784 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 172 T^{2} + 15786 T^{4} - 1010672 T^{6} + 47929043 T^{8} - 1010672 p^{2} T^{10} + 15786 p^{4} T^{12} - 172 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 - 8 T + 32 T^{2} - 368 T^{3} + 1438 T^{4} + 5752 T^{5} - 24320 T^{6} + 270552 T^{7} - 3004701 T^{8} + 270552 p T^{9} - 24320 p^{2} T^{10} + 5752 p^{3} T^{11} + 1438 p^{4} T^{12} - 368 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 - 8 T + 32 T^{2} - 520 T^{3} + 2012 T^{4} + 17432 T^{5} - 68640 T^{6} + 1046296 T^{7} - 15494202 T^{8} + 1046296 p T^{9} - 68640 p^{2} T^{10} + 17432 p^{3} T^{11} + 2012 p^{4} T^{12} - 520 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 + 32 T + 512 T^{2} + 6592 T^{3} + 79454 T^{4} + 821920 T^{5} + 7348224 T^{6} + 61872608 T^{7} + 480643491 T^{8} + 61872608 p T^{9} + 7348224 p^{2} T^{10} + 821920 p^{3} T^{11} + 79454 p^{4} T^{12} + 6592 p^{5} T^{13} + 512 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \)
59 \( ( 1 - 4 T + 170 T^{2} - 856 T^{3} + 13027 T^{4} - 856 p T^{5} + 170 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 16 T + 222 T^{2} + 2480 T^{3} + 20579 T^{4} + 2480 p T^{5} + 222 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 + 16 T + 128 T^{2} + 1648 T^{3} + 12196 T^{4} - 9104 T^{5} - 348800 T^{6} - 6630000 T^{7} - 95097114 T^{8} - 6630000 p T^{9} - 348800 p^{2} T^{10} - 9104 p^{3} T^{11} + 12196 p^{4} T^{12} + 1648 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
71 \( ( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( 1 + 28 p T^{4} + 20460870 T^{8} + 28 p^{5} T^{12} + p^{8} T^{16} \)
79 \( 1 - 320 T^{2} + 58500 T^{4} - 7347136 T^{6} + 669209222 T^{8} - 7347136 p^{2} T^{10} + 58500 p^{4} T^{12} - 320 p^{6} T^{14} + p^{8} T^{16} \)
83 \( 1 - 8 T + 32 T^{2} - 160 T^{3} + 10142 T^{4} - 122968 T^{5} + 672000 T^{6} - 7632056 T^{7} + 75142083 T^{8} - 7632056 p T^{9} + 672000 p^{2} T^{10} - 122968 p^{3} T^{11} + 10142 p^{4} T^{12} - 160 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
89 \( ( 1 + 8 T + 316 T^{2} + 1912 T^{3} + 40422 T^{4} + 1912 p T^{5} + 316 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 + 16 T + 128 T^{2} + 1136 T^{3} + 10556 T^{4} + 116816 T^{5} + 1163136 T^{6} + 11493168 T^{7} + 113239174 T^{8} + 11493168 p T^{9} + 1163136 p^{2} T^{10} + 116816 p^{3} T^{11} + 10556 p^{4} T^{12} + 1136 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.55265849299716031694199798624, −3.50952805597037217430825853028, −3.43368424519402483816116583796, −3.18216560597100365280865005545, −2.96360891159280893940080744495, −2.95552284891903965096566697286, −2.89598299012985863014463256416, −2.72790199250712783389672063311, −2.69069404878822044393461544350, −2.49896099946250439864773525395, −2.48798244637996470348067801365, −2.30114010135297293927146031234, −2.10758038220133630552309515733, −1.94637393172584072472016517687, −1.80628484027739147500064729673, −1.77421653523002986615034036355, −1.65167427656059872788116592166, −1.54160475716783851863535452208, −1.30617622343175463711703285392, −1.22682663815592203146025182891, −0.77592535775976521421220301103, −0.65507076169128815026289976476, −0.59527336924763467924883553242, −0.25124197408795609729630459576, −0.094847808979907444558789489842, 0.094847808979907444558789489842, 0.25124197408795609729630459576, 0.59527336924763467924883553242, 0.65507076169128815026289976476, 0.77592535775976521421220301103, 1.22682663815592203146025182891, 1.30617622343175463711703285392, 1.54160475716783851863535452208, 1.65167427656059872788116592166, 1.77421653523002986615034036355, 1.80628484027739147500064729673, 1.94637393172584072472016517687, 2.10758038220133630552309515733, 2.30114010135297293927146031234, 2.48798244637996470348067801365, 2.49896099946250439864773525395, 2.69069404878822044393461544350, 2.72790199250712783389672063311, 2.89598299012985863014463256416, 2.95552284891903965096566697286, 2.96360891159280893940080744495, 3.18216560597100365280865005545, 3.43368424519402483816116583796, 3.50952805597037217430825853028, 3.55265849299716031694199798624

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

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