Properties

Degree 16
Conductor $ 3^{8} \cdot 5^{8} \cdot 7^{8} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 12·3-s + 7·4-s − 24·6-s − 16·7-s + 6·8-s + 78·9-s + 20·11-s − 84·12-s − 32·14-s + 18·16-s − 18·17-s + 156·18-s + 192·21-s + 40·22-s + 62·23-s − 72·24-s + 10·25-s − 360·27-s − 112·28-s − 100·29-s − 126·31-s − 240·33-s − 36·34-s + 546·36-s − 80·37-s + 384·42-s + ⋯
L(s)  = 1  + 2-s − 4·3-s + 7/4·4-s − 4·6-s − 2.28·7-s + 3/4·8-s + 26/3·9-s + 1.81·11-s − 7·12-s − 2.28·14-s + 9/8·16-s − 1.05·17-s + 26/3·18-s + 64/7·21-s + 1.81·22-s + 2.69·23-s − 3·24-s + 2/5·25-s − 13.3·27-s − 4·28-s − 3.44·29-s − 4.06·31-s − 7.27·33-s − 1.05·34-s + 91/6·36-s − 2.16·37-s + 64/7·42-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(3^{8} \cdot 5^{8} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 3^{8} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.00233628\)
\(L(\frac12)\)  \(\approx\)  \(0.00233628\)
\(L(2)\)   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 \( ( 1 + p T + p T^{2} )^{4} \)
5 \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
7 \( 1 + 16 T + 109 T^{2} + 16 p^{2} T^{3} + 136 p^{2} T^{4} + 16 p^{4} T^{5} + 109 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \)
good2 \( 1 - p T - 3 T^{2} + 7 p T^{3} - 13 T^{4} - 9 p T^{5} + 35 p T^{6} - 11 p^{2} T^{7} - 39 p^{2} T^{8} - 11 p^{4} T^{9} + 35 p^{5} T^{10} - 9 p^{7} T^{11} - 13 p^{8} T^{12} + 7 p^{11} T^{13} - 3 p^{12} T^{14} - p^{15} T^{15} + p^{16} T^{16} \)
11 \( 1 - 20 T - 147 T^{2} + 2960 T^{3} + 57131 T^{4} - 519480 T^{5} - 8882912 T^{6} + 8003440 T^{7} + 1602642534 T^{8} + 8003440 p^{2} T^{9} - 8882912 p^{4} T^{10} - 519480 p^{6} T^{11} + 57131 p^{8} T^{12} + 2960 p^{10} T^{13} - 147 p^{12} T^{14} - 20 p^{14} T^{15} + p^{16} T^{16} \)
13 \( 1 - 188 T^{2} + 39826 T^{4} - 8798048 T^{6} + 2113175419 T^{8} - 8798048 p^{4} T^{10} + 39826 p^{8} T^{12} - 188 p^{12} T^{14} + p^{16} T^{16} \)
17 \( 1 + 18 T + 766 T^{2} + 11844 T^{3} + 262438 T^{4} + 1906254 T^{5} + 22625848 T^{6} - 382636314 T^{7} - 5169040877 T^{8} - 382636314 p^{2} T^{9} + 22625848 p^{4} T^{10} + 1906254 p^{6} T^{11} + 262438 p^{8} T^{12} + 11844 p^{10} T^{13} + 766 p^{12} T^{14} + 18 p^{14} T^{15} + p^{16} T^{16} \)
19 \( 1 + 598 T^{2} + 183481 T^{4} - 682560 T^{5} - 48973562 T^{6} - 501474240 T^{7} - 32269961996 T^{8} - 501474240 p^{2} T^{9} - 48973562 p^{4} T^{10} - 682560 p^{6} T^{11} + 183481 p^{8} T^{12} + 598 p^{12} T^{14} + p^{16} T^{16} \)
23 \( 1 - 62 T + 1497 T^{2} + 6014 T^{3} - 1196893 T^{4} + 31086552 T^{5} - 143771420 T^{6} - 13896561704 T^{7} + 513552019554 T^{8} - 13896561704 p^{2} T^{9} - 143771420 p^{4} T^{10} + 31086552 p^{6} T^{11} - 1196893 p^{8} T^{12} + 6014 p^{10} T^{13} + 1497 p^{12} T^{14} - 62 p^{14} T^{15} + p^{16} T^{16} \)
29 \( ( 1 + 50 T + 1234 T^{2} - 15850 T^{3} - 1164374 T^{4} - 15850 p^{2} T^{5} + 1234 p^{4} T^{6} + 50 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
31 \( 1 + 126 T + 9883 T^{2} + 578466 T^{3} + 27206317 T^{4} + 1079090100 T^{5} + 38160094402 T^{6} + 1243487527488 T^{7} + 38998740329170 T^{8} + 1243487527488 p^{2} T^{9} + 38160094402 p^{4} T^{10} + 1079090100 p^{6} T^{11} + 27206317 p^{8} T^{12} + 578466 p^{10} T^{13} + 9883 p^{12} T^{14} + 126 p^{14} T^{15} + p^{16} T^{16} \)
37 \( 1 + 80 T + 1194 T^{2} + 28960 T^{3} + 3461705 T^{4} - 28416960 T^{5} - 6540374054 T^{6} - 198858748720 T^{7} - 6603314864556 T^{8} - 198858748720 p^{2} T^{9} - 6540374054 p^{4} T^{10} - 28416960 p^{6} T^{11} + 3461705 p^{8} T^{12} + 28960 p^{10} T^{13} + 1194 p^{12} T^{14} + 80 p^{14} T^{15} + p^{16} T^{16} \)
41 \( 1 - 10106 T^{2} + 48877645 T^{4} - 146585251874 T^{6} + 296639674915264 T^{8} - 146585251874 p^{4} T^{10} + 48877645 p^{8} T^{12} - 10106 p^{12} T^{14} + p^{16} T^{16} \)
43 \( ( 1 - 176 T + 17017 T^{2} - 1139948 T^{3} + 56853640 T^{4} - 1139948 p^{2} T^{5} + 17017 p^{4} T^{6} - 176 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
47 \( 1 + 72 T + 233 p T^{2} + 664056 T^{3} + 65519473 T^{4} + 3342900456 T^{5} + 246192812578 T^{6} + 10750018584384 T^{7} + 651041931981118 T^{8} + 10750018584384 p^{2} T^{9} + 246192812578 p^{4} T^{10} + 3342900456 p^{6} T^{11} + 65519473 p^{8} T^{12} + 664056 p^{10} T^{13} + 233 p^{13} T^{14} + 72 p^{14} T^{15} + p^{16} T^{16} \)
53 \( 1 + 76 T - 3069 T^{2} - 443764 T^{3} + 2229785 T^{4} + 1396117872 T^{5} + 34651196266 T^{6} - 1991894657480 T^{7} - 169369280357850 T^{8} - 1991894657480 p^{2} T^{9} + 34651196266 p^{4} T^{10} + 1396117872 p^{6} T^{11} + 2229785 p^{8} T^{12} - 443764 p^{10} T^{13} - 3069 p^{12} T^{14} + 76 p^{14} T^{15} + p^{16} T^{16} \)
59 \( 1 + 54 T + 122 p T^{2} + 336204 T^{3} + 19932742 T^{4} - 202333950 T^{5} - 35478676088 T^{6} - 7641841019598 T^{7} - 385856896323245 T^{8} - 7641841019598 p^{2} T^{9} - 35478676088 p^{4} T^{10} - 202333950 p^{6} T^{11} + 19932742 p^{8} T^{12} + 336204 p^{10} T^{13} + 122 p^{13} T^{14} + 54 p^{14} T^{15} + p^{16} T^{16} \)
61 \( 1 + 396 T + 83164 T^{2} + 12233232 T^{3} + 1413738778 T^{4} + 136250283708 T^{5} + 11318984386192 T^{6} + 825650586150588 T^{7} + 53403008176121923 T^{8} + 825650586150588 p^{2} T^{9} + 11318984386192 p^{4} T^{10} + 136250283708 p^{6} T^{11} + 1413738778 p^{8} T^{12} + 12233232 p^{10} T^{13} + 83164 p^{12} T^{14} + 396 p^{14} T^{15} + p^{16} T^{16} \)
67 \( 1 - 184 T + 93 p T^{2} + 140176 T^{3} + 74370665 T^{4} - 7237038408 T^{5} + 24063966106 T^{6} - 15184872524680 T^{7} + 3087140085953070 T^{8} - 15184872524680 p^{2} T^{9} + 24063966106 p^{4} T^{10} - 7237038408 p^{6} T^{11} + 74370665 p^{8} T^{12} + 140176 p^{10} T^{13} + 93 p^{13} T^{14} - 184 p^{14} T^{15} + p^{16} T^{16} \)
71 \( ( 1 - 82 T + 12166 T^{2} - 846262 T^{3} + 94594474 T^{4} - 846262 p^{2} T^{5} + 12166 p^{4} T^{6} - 82 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
73 \( 1 - 348 T + 68263 T^{2} - 9707460 T^{3} + 1072498525 T^{4} - 96253557984 T^{5} + 7434307414846 T^{6} - 526544361727584 T^{7} + 37365046682274814 T^{8} - 526544361727584 p^{2} T^{9} + 7434307414846 p^{4} T^{10} - 96253557984 p^{6} T^{11} + 1072498525 p^{8} T^{12} - 9707460 p^{10} T^{13} + 68263 p^{12} T^{14} - 348 p^{14} T^{15} + p^{16} T^{16} \)
79 \( 1 + 206 T + 5583 T^{2} - 659438 T^{3} + 124066817 T^{4} + 19494076044 T^{5} + 428008398310 T^{6} + 33090623674568 T^{7} + 7605703397631354 T^{8} + 33090623674568 p^{2} T^{9} + 428008398310 p^{4} T^{10} + 19494076044 p^{6} T^{11} + 124066817 p^{8} T^{12} - 659438 p^{10} T^{13} + 5583 p^{12} T^{14} + 206 p^{14} T^{15} + p^{16} T^{16} \)
83 \( 1 - 20672 T^{2} + 223804480 T^{4} - 2182268545136 T^{6} + 17948924233578718 T^{8} - 2182268545136 p^{4} T^{10} + 223804480 p^{8} T^{12} - 20672 p^{12} T^{14} + p^{16} T^{16} \)
89 \( 1 - 282 T + 59686 T^{2} - 9356196 T^{3} + 1240796086 T^{4} - 138656838366 T^{5} + 14271061565800 T^{6} - 1337157406377822 T^{7} + 121622616146107507 T^{8} - 1337157406377822 p^{2} T^{9} + 14271061565800 p^{4} T^{10} - 138656838366 p^{6} T^{11} + 1240796086 p^{8} T^{12} - 9356196 p^{10} T^{13} + 59686 p^{12} T^{14} - 282 p^{14} T^{15} + p^{16} T^{16} \)
97 \( 1 - 44576 T^{2} + 925514428 T^{4} - 12414040936928 T^{6} + 128325632901816454 T^{8} - 12414040936928 p^{4} T^{10} + 925514428 p^{8} T^{12} - 44576 p^{12} T^{14} + p^{16} T^{16} \)
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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

−6.18021160223615948083412274311, −5.97644962425724822216300478228, −5.81308372160759255613955500692, −5.73685808554343940944775344331, −5.48525760170800922757451307989, −5.43251288816893698292694203459, −5.29500350107647621641387515579, −5.27292227560566692531178341960, −4.88615911383379877913401407471, −4.50951009309382228806057501457, −4.40275203480249749543121459282, −4.39460168875906015145954876621, −3.86134600669325916665439884792, −3.82545665102140960518050881921, −3.66007775418691444447946620530, −3.43670449701224821037597431779, −3.33884572194486714846542008106, −2.89933062494391018806016300811, −2.72573715326276973194961801925, −2.15252034120205693244089681994, −1.90301847333405638741661081241, −1.75432148002059512357107504946, −1.05745444461378650393300367330, −0.833142826775340788140930803211, −0.01654616929462408343970243807, 0.01654616929462408343970243807, 0.833142826775340788140930803211, 1.05745444461378650393300367330, 1.75432148002059512357107504946, 1.90301847333405638741661081241, 2.15252034120205693244089681994, 2.72573715326276973194961801925, 2.89933062494391018806016300811, 3.33884572194486714846542008106, 3.43670449701224821037597431779, 3.66007775418691444447946620530, 3.82545665102140960518050881921, 3.86134600669325916665439884792, 4.39460168875906015145954876621, 4.40275203480249749543121459282, 4.50951009309382228806057501457, 4.88615911383379877913401407471, 5.27292227560566692531178341960, 5.29500350107647621641387515579, 5.43251288816893698292694203459, 5.48525760170800922757451307989, 5.73685808554343940944775344331, 5.81308372160759255613955500692, 5.97644962425724822216300478228, 6.18021160223615948083412274311

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

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