Properties

Degree $16$
Conductor $1.406\times 10^{12}$
Sign $1$
Motivic weight $4$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 26·4-s − 36·5-s + 108·9-s + 36·11-s + 387·16-s − 936·20-s + 516·23-s − 2.99e3·25-s + 2.75e3·31-s + 2.80e3·36-s + 5.29e3·37-s + 936·44-s − 3.88e3·45-s + 420·47-s + 6.18e3·49-s + 3.54e3·53-s − 1.29e3·55-s − 1.66e4·59-s + 3.38e3·64-s − 3.65e3·67-s − 1.32e4·71-s − 1.39e4·80-s + 7.29e3·81-s + 1.55e4·89-s + 1.34e4·92-s + 7.62e3·97-s + 3.88e3·99-s + ⋯
L(s)  = 1  + 13/8·4-s − 1.43·5-s + 4/3·9-s + 0.297·11-s + 1.51·16-s − 2.33·20-s + 0.975·23-s − 4.78·25-s + 2.86·31-s + 13/6·36-s + 3.86·37-s + 0.483·44-s − 1.91·45-s + 0.190·47-s + 2.57·49-s + 1.26·53-s − 0.428·55-s − 4.77·59-s + 0.826·64-s − 0.814·67-s − 2.62·71-s − 2.17·80-s + 10/9·81-s + 1.96·89-s + 1.58·92-s + 0.810·97-s + 0.396·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 11^{8}\)
Sign: $1$
Motivic weight: \(4\)
Character: induced by $\chi_{33} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 11^{8} ,\ ( \ : [2]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(4.00816\)
\(L(\frac12)\) \(\approx\) \(4.00816\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - p^{3} T^{2} )^{4} \)
11 \( 1 - 36 T - 4816 T^{2} - 101940 p T^{3} + 120330 p^{3} T^{4} - 101940 p^{5} T^{5} - 4816 p^{8} T^{6} - 36 p^{12} T^{7} + p^{16} T^{8} \)
good2 \( 1 - 13 p T^{2} + 289 T^{4} - 209 p^{2} T^{6} + 517 p^{2} T^{8} - 209 p^{10} T^{10} + 289 p^{16} T^{12} - 13 p^{25} T^{14} + p^{32} T^{16} \)
5 \( ( 1 + 18 T + 1982 T^{2} + 30438 T^{3} + 1746834 T^{4} + 30438 p^{4} T^{5} + 1982 p^{8} T^{6} + 18 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
7 \( 1 - 884 p T^{2} + 11678572 T^{4} - 930952796 p T^{6} + 1490364730198 T^{8} - 930952796 p^{9} T^{10} + 11678572 p^{16} T^{12} - 884 p^{25} T^{14} + p^{32} T^{16} \)
13 \( 1 - 162008 T^{2} + 12590117968 T^{4} - 619436460996728 T^{6} + 21062441542359634558 T^{8} - 619436460996728 p^{8} T^{10} + 12590117968 p^{16} T^{12} - 162008 p^{24} T^{14} + p^{32} T^{16} \)
17 \( 1 - 341468 T^{2} + 59012100712 T^{4} - 6828905158505300 T^{6} + \)\(62\!\cdots\!46\)\( T^{8} - 6828905158505300 p^{8} T^{10} + 59012100712 p^{16} T^{12} - 341468 p^{24} T^{14} + p^{32} T^{16} \)
19 \( 1 - 30284 p T^{2} + 152808356152 T^{4} - 26685780964652108 T^{6} + \)\(37\!\cdots\!98\)\( T^{8} - 26685780964652108 p^{8} T^{10} + 152808356152 p^{16} T^{12} - 30284 p^{25} T^{14} + p^{32} T^{16} \)
23 \( ( 1 - 258 T + 660806 T^{2} - 61700622 T^{3} + 215285764002 T^{4} - 61700622 p^{4} T^{5} + 660806 p^{8} T^{6} - 258 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
29 \( 1 - 2005364 T^{2} + 2077725355336 T^{4} - 1841644838469189788 T^{6} + \)\(14\!\cdots\!54\)\( T^{8} - 1841644838469189788 p^{8} T^{10} + 2077725355336 p^{16} T^{12} - 2005364 p^{24} T^{14} + p^{32} T^{16} \)
31 \( ( 1 - 1376 T + 3679672 T^{2} - 3640691072 T^{3} + 5096962849966 T^{4} - 3640691072 p^{4} T^{5} + 3679672 p^{8} T^{6} - 1376 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
37 \( ( 1 - 2648 T + 9245032 T^{2} - 15286625576 T^{3} + 27654983547214 T^{4} - 15286625576 p^{4} T^{5} + 9245032 p^{8} T^{6} - 2648 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
41 \( 1 - 3972980 T^{2} + 15101282902792 T^{4} - 53462970682481231900 T^{6} + \)\(21\!\cdots\!90\)\( T^{8} - 53462970682481231900 p^{8} T^{10} + 15101282902792 p^{16} T^{12} - 3972980 p^{24} T^{14} + p^{32} T^{16} \)
43 \( 1 - 18385892 T^{2} + 170175980842552 T^{4} - \)\(10\!\cdots\!76\)\( T^{6} + \)\(40\!\cdots\!14\)\( T^{8} - \)\(10\!\cdots\!76\)\( p^{8} T^{10} + 170175980842552 p^{16} T^{12} - 18385892 p^{24} T^{14} + p^{32} T^{16} \)
47 \( ( 1 - 210 T + 12240722 T^{2} - 2782593654 T^{3} + 84242391941250 T^{4} - 2782593654 p^{4} T^{5} + 12240722 p^{8} T^{6} - 210 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
53 \( ( 1 - 1770 T + 319702 p T^{2} - 10757127678 T^{3} + 127498934985138 T^{4} - 10757127678 p^{4} T^{5} + 319702 p^{9} T^{6} - 1770 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
59 \( ( 1 + 8316 T + 56229104 T^{2} + 271941082596 T^{3} + 17443356249882 p T^{4} + 271941082596 p^{4} T^{5} + 56229104 p^{8} T^{6} + 8316 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
61 \( 1 - 45292376 T^{2} + 1317023575398064 T^{4} - \)\(27\!\cdots\!16\)\( T^{6} + \)\(43\!\cdots\!50\)\( T^{8} - \)\(27\!\cdots\!16\)\( p^{8} T^{10} + 1317023575398064 p^{16} T^{12} - 45292376 p^{24} T^{14} + p^{32} T^{16} \)
67 \( ( 1 + 1828 T + 26966776 T^{2} - 111245917268 T^{3} + 82170182924782 T^{4} - 111245917268 p^{4} T^{5} + 26966776 p^{8} T^{6} + 1828 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
71 \( ( 1 + 6606 T + 54779210 T^{2} + 58180973226 T^{3} + 636995684602866 T^{4} + 58180973226 p^{4} T^{5} + 54779210 p^{8} T^{6} + 6606 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
73 \( 1 - 165039416 T^{2} + 13195111482415036 T^{4} - \)\(65\!\cdots\!96\)\( T^{6} + \)\(22\!\cdots\!74\)\( T^{8} - \)\(65\!\cdots\!96\)\( p^{8} T^{10} + 13195111482415036 p^{16} T^{12} - 165039416 p^{24} T^{14} + p^{32} T^{16} \)
79 \( 1 + 7078036 T^{2} + 3829765928741164 T^{4} + \)\(41\!\cdots\!64\)\( T^{6} + \)\(77\!\cdots\!14\)\( T^{8} + \)\(41\!\cdots\!64\)\( p^{8} T^{10} + 3829765928741164 p^{16} T^{12} + 7078036 p^{24} T^{14} + p^{32} T^{16} \)
83 \( 1 - 135574280 T^{2} + 10334922471866332 T^{4} - \)\(61\!\cdots\!96\)\( T^{6} + \)\(30\!\cdots\!50\)\( T^{8} - \)\(61\!\cdots\!96\)\( p^{8} T^{10} + 10334922471866332 p^{16} T^{12} - 135574280 p^{24} T^{14} + p^{32} T^{16} \)
89 \( ( 1 - 7764 T + 181541324 T^{2} - 755869455948 T^{3} + 13637128342858662 T^{4} - 755869455948 p^{4} T^{5} + 181541324 p^{8} T^{6} - 7764 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
97 \( ( 1 - 3812 T + 232957276 T^{2} - 518258717228 T^{3} + 25105593700183222 T^{4} - 518258717228 p^{4} T^{5} + 232957276 p^{8} T^{6} - 3812 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
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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

−7.39972936833355495526211769984, −7.27442111591566808003601560608, −7.22522899829977447971499077824, −6.92162165236381806645438406085, −6.37336688074956506413783619405, −6.23236901964968190106005010021, −6.20253280418315533846674063426, −5.96618750392371359288711092566, −5.95999018942273204016869174887, −5.64579547093016140168069476546, −5.20318308081748477483932520352, −4.81129686017761205338950136874, −4.39227213821317594808471829869, −4.37054655346817050757255810574, −4.20468863388086682595556654100, −4.09881976656108944280268641503, −3.64317577682149021949781282247, −3.24538360669046602277161269502, −3.04645909623046377122854596137, −2.53207832946159250583868089707, −2.36244205006677585407388664530, −1.89425696227464918216551570556, −1.45791979145342440924391605888, −0.991515655656918157829289301245, −0.41487617480888194126538289288, 0.41487617480888194126538289288, 0.991515655656918157829289301245, 1.45791979145342440924391605888, 1.89425696227464918216551570556, 2.36244205006677585407388664530, 2.53207832946159250583868089707, 3.04645909623046377122854596137, 3.24538360669046602277161269502, 3.64317577682149021949781282247, 4.09881976656108944280268641503, 4.20468863388086682595556654100, 4.37054655346817050757255810574, 4.39227213821317594808471829869, 4.81129686017761205338950136874, 5.20318308081748477483932520352, 5.64579547093016140168069476546, 5.95999018942273204016869174887, 5.96618750392371359288711092566, 6.20253280418315533846674063426, 6.23236901964968190106005010021, 6.37336688074956506413783619405, 6.92162165236381806645438406085, 7.22522899829977447971499077824, 7.27442111591566808003601560608, 7.39972936833355495526211769984

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

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