Properties

Degree 16
Conductor $ 2^{24} \cdot 3^{8} \cdot 5^{16} $
Sign $1$
Motivic weight 1
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 − 8·3-s + 16·6-s + 36·9-s + 6·16-s − 72·18-s − 120·27-s + 8·31-s − 8·32-s − 8·43-s − 48·48-s + 28·49-s + 8·53-s + 240·54-s − 16·62-s + 8·64-s + 24·67-s − 40·71-s + 16·79-s + 330·81-s − 32·83-s + 16·86-s − 64·93-s + 64·96-s − 56·98-s − 16·106-s − 32·107-s + ⋯
L(s)  = 1  − 1.41·2-s − 4.61·3-s + 6.53·6-s + 12·9-s + 3/2·16-s − 16.9·18-s − 23.0·27-s + 1.43·31-s − 1.41·32-s − 1.21·43-s − 6.92·48-s + 4·49-s + 1.09·53-s + 32.6·54-s − 2.03·62-s + 64-s + 2.93·67-s − 4.74·71-s + 1.80·79-s + 36.6·81-s − 3.51·83-s + 1.72·86-s − 6.63·93-s + 6.53·96-s − 5.65·98-s − 1.55·106-s − 3.09·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{24} \cdot 3^{8} \cdot 5^{16}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{600} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 2^{24} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(0.160636\)
\(L(\frac12)\)  \(\approx\)  \(0.160636\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + 5 p T^{4} + p^{4} T^{5} + p^{4} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
3 \( ( 1 + T )^{8} \)
5 \( 1 \)
good7 \( 1 - 4 p T^{2} + 330 T^{4} - 2224 T^{6} + 13203 T^{8} - 2224 p^{2} T^{10} + 330 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \)
11 \( 1 - 56 T^{2} + 1612 T^{4} - 29896 T^{6} + 388998 T^{8} - 29896 p^{2} T^{10} + 1612 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \)
13 \( ( 1 + 30 T^{2} + 32 T^{3} + 451 T^{4} + 32 p T^{5} + 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( 1 - 56 T^{2} + 1484 T^{4} - 24968 T^{6} + 374950 T^{8} - 24968 p^{2} T^{10} + 1484 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \)
19 \( 1 - 36 T^{2} + 1546 T^{4} - 35120 T^{6} + 832243 T^{8} - 35120 p^{2} T^{10} + 1546 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \)
23 \( 1 - 56 T^{2} + 1324 T^{4} - 1592 p T^{6} + 1094310 T^{8} - 1592 p^{3} T^{10} + 1324 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \)
29 \( 1 - 88 T^{2} + 4780 T^{4} - 171048 T^{6} + 5385990 T^{8} - 171048 p^{2} T^{10} + 4780 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \)
31 \( ( 1 - 4 T + 54 T^{2} - 168 T^{3} + 2099 T^{4} - 168 p T^{5} + 54 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 84 T^{2} + 128 T^{3} + 3542 T^{4} + 128 p T^{5} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
41 \( ( 1 + 100 T^{2} + 56 T^{3} + 126 p T^{4} + 56 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 + 4 T + 58 T^{2} + 336 T^{3} + 3379 T^{4} + 336 p T^{5} + 58 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 - 232 T^{2} + 26076 T^{4} - 1910872 T^{6} + 102863686 T^{8} - 1910872 p^{2} T^{10} + 26076 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \)
53 \( ( 1 - 4 T + 92 T^{2} + 44 T^{3} + 3982 T^{4} + 44 p T^{5} + 92 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 - 40 T^{2} + 2364 T^{4} - 112984 T^{6} + 20250598 T^{8} - 112984 p^{2} T^{10} + 2364 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16} \)
61 \( 1 - 252 T^{2} + 32650 T^{4} - 2942672 T^{6} + 202734451 T^{8} - 2942672 p^{2} T^{10} + 32650 p^{4} T^{12} - 252 p^{6} T^{14} + p^{8} T^{16} \)
67 \( ( 1 - 12 T + 154 T^{2} - 1520 T^{3} + 16755 T^{4} - 1520 p T^{5} + 154 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 20 T + 380 T^{2} + 4188 T^{3} + 43342 T^{4} + 4188 p T^{5} + 380 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 - 184 T^{2} + 15196 T^{4} - 1235336 T^{6} + 104948486 T^{8} - 1235336 p^{2} T^{10} + 15196 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - 8 T + 132 T^{2} - 1032 T^{3} + 16454 T^{4} - 1032 p T^{5} + 132 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 16 T + 276 T^{2} + 2576 T^{3} + 30070 T^{4} + 2576 p T^{5} + 276 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 132 T^{2} - 64 T^{3} + 18534 T^{4} - 64 p T^{5} + 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( 1 - 356 T^{2} + 80362 T^{4} - 11927760 T^{6} + 1353370099 T^{8} - 11927760 p^{2} T^{10} + 80362 p^{4} T^{12} - 356 p^{6} T^{14} + p^{8} T^{16} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.99591608274929784726825121237, −4.46645372898650948216136211713, −4.41175275965834762692367783102, −4.22677373175138084900531837283, −4.12944620590399718608812948411, −4.11684940789948159194716655586, −4.05528042263026462920474087232, −3.85735475014325986404458960568, −3.66650105086487649643329325288, −3.43344876378660497944658171183, −3.34857366994614802628597461169, −2.85090310150836847616522388435, −2.84501346881526334164471579589, −2.73613347639555669539913242745, −2.62346636959426063065352770316, −2.31734154628639968216076160824, −1.95201572928151789800875427303, −1.65741592240601726137252813628, −1.63180324251250248226735114995, −1.54654888827323749865831004164, −1.10027247022664126195243664359, −0.799521571847183911392092541634, −0.75378782381264925568275376412, −0.39865249560478206022760976376, −0.38538771242626630338935628688, 0.38538771242626630338935628688, 0.39865249560478206022760976376, 0.75378782381264925568275376412, 0.799521571847183911392092541634, 1.10027247022664126195243664359, 1.54654888827323749865831004164, 1.63180324251250248226735114995, 1.65741592240601726137252813628, 1.95201572928151789800875427303, 2.31734154628639968216076160824, 2.62346636959426063065352770316, 2.73613347639555669539913242745, 2.84501346881526334164471579589, 2.85090310150836847616522388435, 3.34857366994614802628597461169, 3.43344876378660497944658171183, 3.66650105086487649643329325288, 3.85735475014325986404458960568, 4.05528042263026462920474087232, 4.11684940789948159194716655586, 4.12944620590399718608812948411, 4.22677373175138084900531837283, 4.41175275965834762692367783102, 4.46645372898650948216136211713, 4.99591608274929784726825121237

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

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