Properties

Degree 16
Conductor $ 3^{8} \cdot 5^{8} \cdot 7^{8} $
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  − 3·2-s + 3-s + 2·4-s + 4·5-s − 3·6-s + 2·7-s + 3·8-s − 2·9-s − 12·10-s + 2·12-s − 6·14-s + 4·15-s − 3·16-s − 12·17-s + 6·18-s + 9·19-s + 8·20-s + 2·21-s + 27·23-s + 3·24-s + 6·25-s − 27-s + 4·28-s − 12·30-s − 21·31-s + 36·34-s + 8·35-s + ⋯
L(s)  = 1  − 2.12·2-s + 0.577·3-s + 4-s + 1.78·5-s − 1.22·6-s + 0.755·7-s + 1.06·8-s − 2/3·9-s − 3.79·10-s + 0.577·12-s − 1.60·14-s + 1.03·15-s − 3/4·16-s − 2.91·17-s + 1.41·18-s + 2.06·19-s + 1.78·20-s + 0.436·21-s + 5.62·23-s + 0.612·24-s + 6/5·25-s − 0.192·27-s + 0.755·28-s − 2.19·30-s − 3.77·31-s + 6.17·34-s + 1.35·35-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(3^{8} \cdot 5^{8} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 3^{8} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $0.415057$
$L(\frac12)$  $\approx$  $0.415057$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad3 \( 1 - T + p T^{2} - 4 T^{3} + 16 T^{4} - 4 p T^{5} + p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
5 \( ( 1 - T + T^{2} )^{4} \)
7 \( 1 - 2 T + 4 T^{2} + 10 T^{3} - 41 T^{4} + 10 p T^{5} + 4 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
good2 \( 1 + 3 T + 7 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} + 3 p^{2} T^{5} - p^{3} T^{6} - 9 p^{2} T^{7} - 17 p^{2} T^{8} - 9 p^{3} T^{9} - p^{5} T^{10} + 3 p^{5} T^{11} + p^{8} T^{12} + 3 p^{7} T^{13} + 7 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 + 16 T^{2} - 2 T^{4} + 30 T^{5} + 268 T^{6} + 1548 T^{7} + 21079 T^{8} + 1548 p T^{9} + 268 p^{2} T^{10} + 30 p^{3} T^{11} - 2 p^{4} T^{12} + 16 p^{6} T^{14} + p^{8} T^{16} \)
13 \( 1 - 83 T^{2} + 3217 T^{4} - 76058 T^{6} + 1197778 T^{8} - 76058 p^{2} T^{10} + 3217 p^{4} T^{12} - 83 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 + 12 T + 2 p T^{2} + 12 T^{3} + 1078 T^{4} + 6882 T^{5} + 8740 T^{6} + 70272 T^{7} + 637627 T^{8} + 70272 p T^{9} + 8740 p^{2} T^{10} + 6882 p^{3} T^{11} + 1078 p^{4} T^{12} + 12 p^{5} T^{13} + 2 p^{7} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 - 9 T + 100 T^{2} - 657 T^{3} + 4723 T^{4} - 26244 T^{5} + 148996 T^{6} - 704196 T^{7} + 3331528 T^{8} - 704196 p T^{9} + 148996 p^{2} T^{10} - 26244 p^{3} T^{11} + 4723 p^{4} T^{12} - 657 p^{5} T^{13} + 100 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 27 T + 17 p T^{2} - 3996 T^{3} + 31651 T^{4} - 205875 T^{5} + 1157938 T^{6} - 5917779 T^{7} + 28782226 T^{8} - 5917779 p T^{9} + 1157938 p^{2} T^{10} - 205875 p^{3} T^{11} + 31651 p^{4} T^{12} - 3996 p^{5} T^{13} + 17 p^{7} T^{14} - 27 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 53 T^{2} + 3250 T^{4} - 128951 T^{6} + 4063174 T^{8} - 128951 p^{2} T^{10} + 3250 p^{4} T^{12} - 53 p^{6} T^{14} + p^{8} T^{16} \)
31 \( 1 + 21 T + 262 T^{2} + 2415 T^{3} + 17293 T^{4} + 101304 T^{5} + 505090 T^{6} + 2328618 T^{7} + 11769748 T^{8} + 2328618 p T^{9} + 505090 p^{2} T^{10} + 101304 p^{3} T^{11} + 17293 p^{4} T^{12} + 2415 p^{5} T^{13} + 262 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 - 7 T - 24 T^{2} + 493 T^{3} - 973 T^{4} - 16188 T^{5} + 118336 T^{6} + 258098 T^{7} - 5756772 T^{8} + 258098 p T^{9} + 118336 p^{2} T^{10} - 16188 p^{3} T^{11} - 973 p^{4} T^{12} + 493 p^{5} T^{13} - 24 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \)
41 \( ( 1 + 15 T + 218 T^{2} + 1791 T^{3} + 14136 T^{4} + 1791 p T^{5} + 218 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 8 T + 184 T^{2} - 1022 T^{3} + 12127 T^{4} - 1022 p T^{5} + 184 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 + 6 T - 116 T^{2} - 252 T^{3} + 10126 T^{4} + 1986 T^{5} - 595736 T^{6} - 157218 T^{7} + 25623007 T^{8} - 157218 p T^{9} - 595736 p^{2} T^{10} + 1986 p^{3} T^{11} + 10126 p^{4} T^{12} - 252 p^{5} T^{13} - 116 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 24 T + 340 T^{2} - 3552 T^{3} + 29050 T^{4} - 180120 T^{5} + 750160 T^{6} - 1659096 T^{7} + 1273315 T^{8} - 1659096 p T^{9} + 750160 p^{2} T^{10} - 180120 p^{3} T^{11} + 29050 p^{4} T^{12} - 3552 p^{5} T^{13} + 340 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 + 12 T - 80 T^{2} - 1164 T^{3} + 7690 T^{4} + 80082 T^{5} - 434420 T^{6} - 1772232 T^{7} + 28861927 T^{8} - 1772232 p T^{9} - 434420 p^{2} T^{10} + 80082 p^{3} T^{11} + 7690 p^{4} T^{12} - 1164 p^{5} T^{13} - 80 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 - 15 T + 223 T^{2} - 2220 T^{3} + 19711 T^{4} - 141723 T^{5} + 816310 T^{6} - 5175267 T^{7} + 31433836 T^{8} - 5175267 p T^{9} + 816310 p^{2} T^{10} - 141723 p^{3} T^{11} + 19711 p^{4} T^{12} - 2220 p^{5} T^{13} + 223 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 - 4 T - 234 T^{2} + 412 T^{3} + 35255 T^{4} - 28434 T^{5} - 3551522 T^{6} + 717722 T^{7} + 271900824 T^{8} + 717722 p T^{9} - 3551522 p^{2} T^{10} - 28434 p^{3} T^{11} + 35255 p^{4} T^{12} + 412 p^{5} T^{13} - 234 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 464 T^{2} + 99532 T^{4} - 12936548 T^{6} + 1114829374 T^{8} - 12936548 p^{2} T^{10} + 99532 p^{4} T^{12} - 464 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 - 15 T + 280 T^{2} - 3075 T^{3} + 38779 T^{4} - 422928 T^{5} + 4017052 T^{6} - 39506334 T^{7} + 310273396 T^{8} - 39506334 p T^{9} + 4017052 p^{2} T^{10} - 422928 p^{3} T^{11} + 38779 p^{4} T^{12} - 3075 p^{5} T^{13} + 280 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 + 29 T + 294 T^{2} + 25 p T^{3} + 27377 T^{4} + 260496 T^{5} + 598654 T^{6} + 2403434 T^{7} + 77714340 T^{8} + 2403434 p T^{9} + 598654 p^{2} T^{10} + 260496 p^{3} T^{11} + 27377 p^{4} T^{12} + 25 p^{6} T^{13} + 294 p^{6} T^{14} + 29 p^{7} T^{15} + p^{8} T^{16} \)
83 \( ( 1 - 15 T + 380 T^{2} - 3759 T^{3} + 49260 T^{4} - 3759 p T^{5} + 380 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( 1 + 3 T - 53 T^{2} + 2820 T^{3} + 14227 T^{4} - 160275 T^{5} + 3467116 T^{6} + 26593569 T^{7} - 193500020 T^{8} + 26593569 p T^{9} + 3467116 p^{2} T^{10} - 160275 p^{3} T^{11} + 14227 p^{4} T^{12} + 2820 p^{5} T^{13} - 53 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 - 368 T^{2} + 81676 T^{4} - 12257504 T^{6} + 1385094598 T^{8} - 12257504 p^{2} T^{10} + 81676 p^{4} T^{12} - 368 p^{6} T^{14} + p^{8} T^{16} \)
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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

−6.59157549360708681307352304586, −6.52350401469066907492064883978, −6.42635432795906055589815040963, −5.97281425600697849800202054507, −5.57320561660966983847811457706, −5.54767995398334892871223679998, −5.32056021840744485747838014687, −5.29668981685163018397146649296, −5.19942907610908329166890030836, −5.16637667998238812178339920386, −4.99580413979799486386904975575, −4.41423813479039682933611043540, −4.39193814066225622794278350050, −4.12623637892311151497127713046, −3.99637367614305659200120332665, −3.58139211027403480765315332422, −3.20195871473636375834775400327, −3.18813773818817911676988456048, −2.93940084751749054223419473568, −2.61578888808864390979676742143, −2.43595711252732681082797155052, −2.03334558200098910459550346423, −1.76990431563914865496509141169, −1.26368942323815584386174698163, −1.00903400038061717024357276711, 1.00903400038061717024357276711, 1.26368942323815584386174698163, 1.76990431563914865496509141169, 2.03334558200098910459550346423, 2.43595711252732681082797155052, 2.61578888808864390979676742143, 2.93940084751749054223419473568, 3.18813773818817911676988456048, 3.20195871473636375834775400327, 3.58139211027403480765315332422, 3.99637367614305659200120332665, 4.12623637892311151497127713046, 4.39193814066225622794278350050, 4.41423813479039682933611043540, 4.99580413979799486386904975575, 5.16637667998238812178339920386, 5.19942907610908329166890030836, 5.29668981685163018397146649296, 5.32056021840744485747838014687, 5.54767995398334892871223679998, 5.57320561660966983847811457706, 5.97281425600697849800202054507, 6.42635432795906055589815040963, 6.52350401469066907492064883978, 6.59157549360708681307352304586

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

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