Properties

Label 16-546e8-1.1-c1e8-0-17
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 + 4·11-s − 144·12-s + 3·13-s + 24·14-s − 8·15-s + 330·16-s + 4·17-s + 48·18-s − 4·19-s + 72·20-s − 12·21-s + 32·22-s − 8·23-s − 480·24-s + 13·25-s + 24·26-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 + 1.20·11-s − 41.5·12-s + 0.832·13-s + 6.41·14-s − 2.06·15-s + 82.5·16-s + 0.970·17-s + 11.3·18-s − 0.917·19-s + 16.0·20-s − 2.61·21-s + 6.82·22-s − 1.66·23-s − 97.9·24-s + 13/5·25-s + 4.70·26-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

\(L(1)\) \(\approx\) \(287.7656413\)
\(L(\frac12)\) \(\approx\) \(287.7656413\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + 20 T^{2} - 15 T^{3} + 231 T^{4} - 15 p T^{5} + 20 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
good5 \( ( 1 - 7 T + 22 T^{2} - 37 T^{3} + 57 T^{4} - 37 p T^{5} + 22 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )( 1 + p T + 4 T^{2} - 19 T^{3} - 63 T^{4} - 19 p T^{5} + 4 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} ) \)
11 \( 1 - 4 T - 14 T^{2} + 162 T^{3} - 81 T^{4} - 2445 T^{5} + 8427 T^{6} + 14419 T^{7} - 125751 T^{8} + 14419 p T^{9} + 8427 p^{2} T^{10} - 2445 p^{3} T^{11} - 81 p^{4} T^{12} + 162 p^{5} T^{13} - 14 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
17 \( ( 1 - 2 T + 24 T^{2} + 3 p T^{3} + 211 T^{4} + 3 p^{2} T^{5} + 24 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( 1 + 4 T - p T^{2} + 104 T^{3} + 698 T^{4} - 2668 T^{5} + 8301 T^{6} + 62226 T^{7} - 152189 T^{8} + 62226 p T^{9} + 8301 p^{2} T^{10} - 2668 p^{3} T^{11} + 698 p^{4} T^{12} + 104 p^{5} T^{13} - p^{7} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
23 \( ( 1 + 4 T + 58 T^{2} + 195 T^{3} + 1613 T^{4} + 195 p T^{5} + 58 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( 1 - 2 T - 20 T^{2} - 162 T^{3} - 79 T^{4} + 6097 T^{5} + 28467 T^{6} - 124657 T^{7} - 499445 T^{8} - 124657 p T^{9} + 28467 p^{2} T^{10} + 6097 p^{3} T^{11} - 79 p^{4} T^{12} - 162 p^{5} T^{13} - 20 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 - 14 T + 44 T^{2} + 2 p T^{3} + 741 T^{4} - 3461 T^{5} - 53077 T^{6} + 156063 T^{7} + 844921 T^{8} + 156063 p T^{9} - 53077 p^{2} T^{10} - 3461 p^{3} T^{11} + 741 p^{4} T^{12} + 2 p^{6} T^{13} + 44 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \)
37 \( ( 1 - 6 T + 156 T^{2} - 665 T^{3} + 8805 T^{4} - 665 p T^{5} + 156 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 - 12 T - 6 T^{2} + 306 T^{3} + 1865 T^{4} - 9147 T^{5} - 104817 T^{6} + 676101 T^{7} - 2276181 T^{8} + 676101 p T^{9} - 104817 p^{2} T^{10} - 9147 p^{3} T^{11} + 1865 p^{4} T^{12} + 306 p^{5} T^{13} - 6 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 - 138 T^{2} + 146 T^{3} + 10887 T^{4} - 13213 T^{5} - 610013 T^{6} + 270903 T^{7} + 27676345 T^{8} + 270903 p T^{9} - 610013 p^{2} T^{10} - 13213 p^{3} T^{11} + 10887 p^{4} T^{12} + 146 p^{5} T^{13} - 138 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 7 T - 107 T^{2} + 654 T^{3} + 8042 T^{4} - 34657 T^{5} - 424428 T^{6} + 772924 T^{7} + 19410175 T^{8} + 772924 p T^{9} - 424428 p^{2} T^{10} - 34657 p^{3} T^{11} + 8042 p^{4} T^{12} + 654 p^{5} T^{13} - 107 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 + T - 65 T^{2} - 1080 T^{3} + 894 T^{4} + 62409 T^{5} + 478074 T^{6} - 2019040 T^{7} - 28883847 T^{8} - 2019040 p T^{9} + 478074 p^{2} T^{10} + 62409 p^{3} T^{11} + 894 p^{4} T^{12} - 1080 p^{5} T^{13} - 65 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \)
59 \( ( 1 + 16 T + 253 T^{2} + 2356 T^{3} + 21813 T^{4} + 2356 p T^{5} + 253 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( 1 + 4 T - 52 T^{2} + 1256 T^{3} + 6170 T^{4} - 66844 T^{5} + 839472 T^{6} + 5794044 T^{7} - 38376173 T^{8} + 5794044 p T^{9} + 839472 p^{2} T^{10} - 66844 p^{3} T^{11} + 6170 p^{4} T^{12} + 1256 p^{5} T^{13} - 52 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 - 19 T + 76 T^{2} + 197 T^{3} + 2400 T^{4} - 18294 T^{5} - 410307 T^{6} + 5554842 T^{7} - 44341751 T^{8} + 5554842 p T^{9} - 410307 p^{2} T^{10} - 18294 p^{3} T^{11} + 2400 p^{4} T^{12} + 197 p^{5} T^{13} + 76 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 20 T + 151 T^{2} - 948 T^{3} + 5336 T^{4} + 8680 T^{5} - 357951 T^{6} + 6067808 T^{7} - 73033073 T^{8} + 6067808 p T^{9} - 357951 p^{2} T^{10} + 8680 p^{3} T^{11} + 5336 p^{4} T^{12} - 948 p^{5} T^{13} + 151 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 + 7 T - 176 T^{2} - 557 T^{3} + 21594 T^{4} + 15702 T^{5} - 1863345 T^{6} - 909252 T^{7} + 122212573 T^{8} - 909252 p T^{9} - 1863345 p^{2} T^{10} + 15702 p^{3} T^{11} + 21594 p^{4} T^{12} - 557 p^{5} T^{13} - 176 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 24 T + 189 T^{2} - 124 T^{3} - 6786 T^{4} + 100964 T^{5} - 891383 T^{6} - 39114 p T^{7} + 106087891 T^{8} - 39114 p^{2} T^{9} - 891383 p^{2} T^{10} + 100964 p^{3} T^{11} - 6786 p^{4} T^{12} - 124 p^{5} T^{13} + 189 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
83 \( ( 1 + 32 T + 696 T^{2} + 9651 T^{3} + 104187 T^{4} + 9651 p T^{5} + 696 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 11 T + 252 T^{2} - 2433 T^{3} + 31963 T^{4} - 2433 p T^{5} + 252 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 11 T - 187 T^{2} + 962 T^{3} + 32064 T^{4} - 25625 T^{5} - 3767656 T^{6} + 3490380 T^{7} + 292157209 T^{8} + 3490380 p T^{9} - 3767656 p^{2} T^{10} - 25625 p^{3} T^{11} + 32064 p^{4} T^{12} + 962 p^{5} T^{13} - 187 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \)
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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

−4.78792031049237203251871724987, −4.61785409446431762100936562007, −4.33017157196192280283613355380, −4.26187232002925049303997346899, −4.21559989987828443064591951247, −4.13780680459462548692319863473, −4.07881470209480537671837244214, −4.07405097994514655746087476318, −4.02767022098173651880493993052, −3.44463184114609447712093092399, −3.30973445519537675043719688581, −3.13723280502099414913129093500, −3.01428667893019982989718494949, −2.98498107042203867014697177399, −2.82370342219578301511791916773, −2.59630183762964731391340034771, −2.58099667236750177719285108712, −2.15184368730451333108424620849, −1.99558881640034927576938225985, −1.91268365663615367596141253890, −1.76861868489595384093624899460, −1.26343247686759808623242967764, −1.13218417911323628651200122569, −1.02161635613942835727665879266, −0.892733128753621187788149517683, 0.892733128753621187788149517683, 1.02161635613942835727665879266, 1.13218417911323628651200122569, 1.26343247686759808623242967764, 1.76861868489595384093624899460, 1.91268365663615367596141253890, 1.99558881640034927576938225985, 2.15184368730451333108424620849, 2.58099667236750177719285108712, 2.59630183762964731391340034771, 2.82370342219578301511791916773, 2.98498107042203867014697177399, 3.01428667893019982989718494949, 3.13723280502099414913129093500, 3.30973445519537675043719688581, 3.44463184114609447712093092399, 4.02767022098173651880493993052, 4.07405097994514655746087476318, 4.07881470209480537671837244214, 4.13780680459462548692319863473, 4.21559989987828443064591951247, 4.26187232002925049303997346899, 4.33017157196192280283613355380, 4.61785409446431762100936562007, 4.78792031049237203251871724987

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

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