Properties

Label 16-546e8-1.1-c1e8-0-15
Degree $16$
Conductor $7.898\times 10^{21}$
Sign $1$
Analytic cond. $130544.$
Root an. cond. $2.08802$
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  + 8·2-s + 4·3-s + 36·4-s + 2·5-s + 32·6-s − 3·7-s + 120·8-s + 6·9-s + 16·10-s − 6·11-s + 144·12-s − 11·13-s − 24·14-s + 8·15-s + 330·16-s − 8·17-s + 48·18-s + 6·19-s + 72·20-s − 12·21-s − 48·22-s + 20·23-s + 480·24-s + 3·25-s − 88·26-s − 108·28-s + 2·29-s + ⋯
L(s)  = 1  + 5.65·2-s + 2.30·3-s + 18·4-s + 0.894·5-s + 13.0·6-s − 1.13·7-s + 42.4·8-s + 2·9-s + 5.05·10-s − 1.80·11-s + 41.5·12-s − 3.05·13-s − 6.41·14-s + 2.06·15-s + 82.5·16-s − 1.94·17-s + 11.3·18-s + 1.37·19-s + 16.0·20-s − 2.61·21-s − 10.2·22-s + 4.17·23-s + 97.9·24-s + 3/5·25-s − 17.2·26-s − 20.4·28-s + 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(130544.\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{546} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(670.4020618\)
\(L(\frac12)\) \(\approx\) \(670.4020618\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{8} \)
3 \( ( 1 - T + T^{2} )^{4} \)
7 \( 1 + 3 T + 2 T^{2} - 3 p T^{3} - 117 T^{4} - 3 p^{2} T^{5} + 2 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
13 \( 1 + 11 T + 62 T^{2} + 267 T^{3} + 1031 T^{4} + 267 p T^{5} + 62 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
good5 \( 1 - 2 T + T^{2} - 6 T^{3} + 12 T^{4} - 12 T^{5} + 159 T^{6} - 16 p^{2} T^{7} + 159 T^{8} - 16 p^{3} T^{9} + 159 p^{2} T^{10} - 12 p^{3} T^{11} + 12 p^{4} T^{12} - 6 p^{5} T^{13} + p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 + 6 T + 2 T^{2} - 14 T^{3} + 63 T^{4} - 3 p^{2} T^{5} - 2011 T^{6} - 1011 T^{7} + 3345 T^{8} - 1011 p T^{9} - 2011 p^{2} T^{10} - 3 p^{5} T^{11} + 63 p^{4} T^{12} - 14 p^{5} T^{13} + 2 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \)
17 \( ( 1 + 4 T + 54 T^{2} + 213 T^{3} + 1257 T^{4} + 213 p T^{5} + 54 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( 1 - 6 T - 45 T^{2} + 166 T^{3} + 2262 T^{4} - 236 p T^{5} - 62189 T^{6} + 23010 T^{7} + 1465411 T^{8} + 23010 p T^{9} - 62189 p^{2} T^{10} - 236 p^{4} T^{11} + 2262 p^{4} T^{12} + 166 p^{5} T^{13} - 45 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
23 \( ( 1 - 10 T + 82 T^{2} - 535 T^{3} + 2547 T^{4} - 535 p T^{5} + 82 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( 1 - 2 T - 62 T^{2} + 438 T^{3} + 1707 T^{4} - 18759 T^{5} + 38493 T^{6} + 338675 T^{7} - 2208849 T^{8} + 338675 p T^{9} + 38493 p^{2} T^{10} - 18759 p^{3} T^{11} + 1707 p^{4} T^{12} + 438 p^{5} T^{13} - 62 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 - 6 T - 40 T^{2} + 498 T^{3} - 335 T^{4} - 12051 T^{5} + 37955 T^{6} + 80889 T^{7} - 826703 T^{8} + 80889 p T^{9} + 37955 p^{2} T^{10} - 12051 p^{3} T^{11} - 335 p^{4} T^{12} + 498 p^{5} T^{13} - 40 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
37 \( ( 1 - 28 T + 408 T^{2} - 3931 T^{3} + 27665 T^{4} - 3931 p T^{5} + 408 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 - 72 T^{2} + 198 T^{3} + 1259 T^{4} - 11187 T^{5} - 30735 T^{6} + 195921 T^{7} + 3463719 T^{8} + 195921 p T^{9} - 30735 p^{2} T^{10} - 11187 p^{3} T^{11} + 1259 p^{4} T^{12} + 198 p^{5} T^{13} - 72 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 + 6 T - 98 T^{2} - 18 p T^{3} + 5329 T^{4} + 44937 T^{5} - 143171 T^{6} - 901221 T^{7} + 4636105 T^{8} - 901221 p T^{9} - 143171 p^{2} T^{10} + 44937 p^{3} T^{11} + 5329 p^{4} T^{12} - 18 p^{6} T^{13} - 98 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 - T - 131 T^{2} - 78 T^{3} + 9356 T^{4} + 11249 T^{5} - 478776 T^{6} - 277502 T^{7} + 21461767 T^{8} - 277502 p T^{9} - 478776 p^{2} T^{10} + 11249 p^{3} T^{11} + 9356 p^{4} T^{12} - 78 p^{5} T^{13} - 131 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 7 T - 119 T^{2} + 1086 T^{3} + 8036 T^{4} - 78157 T^{5} - 258144 T^{6} + 1916824 T^{7} + 10200157 T^{8} + 1916824 p T^{9} - 258144 p^{2} T^{10} - 78157 p^{3} T^{11} + 8036 p^{4} T^{12} + 1086 p^{5} T^{13} - 119 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \)
59 \( ( 1 + 2 T + 93 T^{2} - 306 T^{3} + 4323 T^{4} - 306 p T^{5} + 93 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( 1 - 24 T + 196 T^{2} - 1248 T^{3} + 15194 T^{4} - 68160 T^{5} - 644208 T^{6} + 6345384 T^{7} - 27687421 T^{8} + 6345384 p T^{9} - 644208 p^{2} T^{10} - 68160 p^{3} T^{11} + 15194 p^{4} T^{12} - 1248 p^{5} T^{13} + 196 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 + 15 T - 72 T^{2} - 1177 T^{3} + 14982 T^{4} + 95414 T^{5} - 1436897 T^{6} - 301836 T^{7} + 148115845 T^{8} - 301836 p T^{9} - 1436897 p^{2} T^{10} + 95414 p^{3} T^{11} + 14982 p^{4} T^{12} - 1177 p^{5} T^{13} - 72 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 6 T - 153 T^{2} + 198 T^{3} + 15790 T^{4} + 18060 T^{5} - 946377 T^{6} - 602562 T^{7} + 42720099 T^{8} - 602562 p T^{9} - 946377 p^{2} T^{10} + 18060 p^{3} T^{11} + 15790 p^{4} T^{12} + 198 p^{5} T^{13} - 153 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 - T - 284 T^{2} + 143 T^{3} + 50058 T^{4} - 14706 T^{5} - 5753901 T^{6} + 372816 T^{7} + 494276869 T^{8} + 372816 p T^{9} - 5753901 p^{2} T^{10} - 14706 p^{3} T^{11} + 50058 p^{4} T^{12} + 143 p^{5} T^{13} - 284 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 + 12 T - 95 T^{2} - 1656 T^{3} + 26 p T^{4} + 49164 T^{5} - 718131 T^{6} + 1559838 T^{7} + 122032043 T^{8} + 1559838 p T^{9} - 718131 p^{2} T^{10} + 49164 p^{3} T^{11} + 26 p^{5} T^{12} - 1656 p^{5} T^{13} - 95 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
83 \( ( 1 + 16 T + 246 T^{2} + 2655 T^{3} + 28977 T^{4} + 2655 p T^{5} + 246 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 25 T + 526 T^{2} + 6855 T^{3} + 77813 T^{4} + 6855 p T^{5} + 526 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 + T - 199 T^{2} - 634 T^{3} + 12828 T^{4} + 70339 T^{5} - 1453960 T^{6} - 2228292 T^{7} + 235898857 T^{8} - 2228292 p T^{9} - 1453960 p^{2} T^{10} + 70339 p^{3} T^{11} + 12828 p^{4} T^{12} - 634 p^{5} T^{13} - 199 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \)
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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

−4.79398177847373631261426592140, −4.50917239324284242106705905436, −4.32745435039998707366762329490, −4.31328749991812180600579542339, −4.27595616495077692720184657183, −4.10909715035425819767296493703, −3.96952982485927793554815017021, −3.90404185854603470782363961804, −3.69503428115492167533167241869, −3.14721144183630960173121111191, −3.04025863020171689780142006450, −3.02494871357376634597234412654, −2.97650281298064673523593771498, −2.96011895992169753206188018258, −2.78642900368569589272471506941, −2.62906258487138382258670146514, −2.52873782507111515886733830617, −2.50219283918103932188771879971, −2.45696151963468978573467466366, −2.11600551289411832966245222616, −2.09574131163699374491228031648, −1.52684578173480360360349497596, −1.22026042075076759750175896581, −1.16357820883724181801072898118, −0.72278464043229839220151938478, 0.72278464043229839220151938478, 1.16357820883724181801072898118, 1.22026042075076759750175896581, 1.52684578173480360360349497596, 2.09574131163699374491228031648, 2.11600551289411832966245222616, 2.45696151963468978573467466366, 2.50219283918103932188771879971, 2.52873782507111515886733830617, 2.62906258487138382258670146514, 2.78642900368569589272471506941, 2.96011895992169753206188018258, 2.97650281298064673523593771498, 3.02494871357376634597234412654, 3.04025863020171689780142006450, 3.14721144183630960173121111191, 3.69503428115492167533167241869, 3.90404185854603470782363961804, 3.96952982485927793554815017021, 4.10909715035425819767296493703, 4.27595616495077692720184657183, 4.31328749991812180600579542339, 4.32745435039998707366762329490, 4.50917239324284242106705905436, 4.79398177847373631261426592140

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

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