Properties

Degree 16
Conductor $ 2^{24} \cdot 7^{16} $
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  + 8·3-s + 6·4-s + 56·9-s − 16·11-s + 48·12-s + 16·16-s + 72·17-s + 8·19-s − 16·25-s + 368·27-s − 128·33-s + 336·36-s + 80·41-s − 160·43-s − 96·44-s + 128·48-s + 576·51-s + 64·57-s + 184·59-s + 72·64-s + 224·67-s + 432·68-s + 232·73-s − 128·75-s + 48·76-s + 1.93e3·81-s − 176·83-s + ⋯
L(s)  = 1  + 8/3·3-s + 3/2·4-s + 56/9·9-s − 1.45·11-s + 4·12-s + 16-s + 4.23·17-s + 8/19·19-s − 0.639·25-s + 13.6·27-s − 3.87·33-s + 28/3·36-s + 1.95·41-s − 3.72·43-s − 2.18·44-s + 8/3·48-s + 11.2·51-s + 1.12·57-s + 3.11·59-s + 9/8·64-s + 3.34·67-s + 6.35·68-s + 3.17·73-s − 1.70·75-s + 0.631·76-s + 23.9·81-s − 2.12·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{24} \cdot 7^{16}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{392} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 2^{24} \cdot 7^{16} ,\ ( \ : [1]^{8} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(141.851\)
\(L(\frac12)\)  \(\approx\)  \(141.851\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 - 3 p T^{2} + 5 p^{2} T^{4} - 3 p^{5} T^{6} + p^{8} T^{8} \)
7 \( 1 \)
good3 \( ( 1 - 4 T - 4 T^{2} - 8 T^{3} + 175 T^{4} - 8 p^{2} T^{5} - 4 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
5 \( 1 + 16 T^{2} + 102 p T^{4} - 24064 T^{6} - 486109 T^{8} - 24064 p^{4} T^{10} + 102 p^{9} T^{12} + 16 p^{12} T^{14} + p^{16} T^{16} \)
11 \( ( 1 + 8 T - 122 T^{2} - 448 T^{3} + 12211 T^{4} - 448 p^{2} T^{5} - 122 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
13 \( ( 1 - 592 T^{2} + 143170 T^{4} - 592 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
17 \( ( 1 - 36 T + 426 T^{2} - 10512 T^{3} + 298835 T^{4} - 10512 p^{2} T^{5} + 426 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
19 \( ( 1 - 4 T + 12 T^{2} + 2872 T^{3} - 136081 T^{4} + 2872 p^{2} T^{5} + 12 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
23 \( 1 + 268 T^{2} - 405462 T^{4} - 22082128 T^{6} + 129391640531 T^{8} - 22082128 p^{4} T^{10} - 405462 p^{8} T^{12} + 268 p^{12} T^{14} + p^{16} T^{16} \)
29 \( ( 1 - 1178 T^{2} + p^{4} T^{4} )^{4} \)
31 \( 1 + 1380 T^{2} + 484426 T^{4} - 589353840 T^{6} - 595318269165 T^{8} - 589353840 p^{4} T^{10} + 484426 p^{8} T^{12} + 1380 p^{12} T^{14} + p^{16} T^{16} \)
37 \( 1 + 1780 T^{2} + 1136778 T^{4} - 3055726000 T^{6} - 5323083215437 T^{8} - 3055726000 p^{4} T^{10} + 1136778 p^{8} T^{12} + 1780 p^{12} T^{14} + p^{16} T^{16} \)
41 \( ( 1 - 20 T + 3174 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
43 \( ( 1 + 40 T + 4090 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
47 \( 1 + 7492 T^{2} + 32739594 T^{4} + 102124261136 T^{6} + 249405993329555 T^{8} + 102124261136 p^{4} T^{10} + 32739594 p^{8} T^{12} + 7492 p^{12} T^{14} + p^{16} T^{16} \)
53 \( 1 + 1604 T^{2} + 8727850 T^{4} - 35185337584 T^{6} - 44676002634701 T^{8} - 35185337584 p^{4} T^{10} + 8727850 p^{8} T^{12} + 1604 p^{12} T^{14} + p^{16} T^{16} \)
59 \( ( 1 - 92 T - 372 T^{2} - 172408 T^{3} + 36494351 T^{4} - 172408 p^{2} T^{5} - 372 p^{4} T^{6} - 92 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
61 \( 1 + 13232 T^{2} + 103975486 T^{4} + 574515656192 T^{6} + 2440762377018979 T^{8} + 574515656192 p^{4} T^{10} + 103975486 p^{8} T^{12} + 13232 p^{12} T^{14} + p^{16} T^{16} \)
67 \( ( 1 - 112 T + 942 T^{2} - 293888 T^{3} + 58145267 T^{4} - 293888 p^{2} T^{5} + 942 p^{4} T^{6} - 112 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
71 \( ( 1 - 9412 T^{2} + 47279686 T^{4} - 9412 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
73 \( ( 1 - 116 T - 6 p T^{2} - 375376 T^{3} + 92937971 T^{4} - 375376 p^{2} T^{5} - 6 p^{5} T^{6} - 116 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( 1 + 16900 T^{2} + 142729866 T^{4} + 1098161526800 T^{6} + 7773091247541395 T^{8} + 1098161526800 p^{4} T^{10} + 142729866 p^{8} T^{12} + 16900 p^{12} T^{14} + p^{16} T^{16} \)
83 \( ( 1 + 44 T + 13924 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
89 \( ( 1 - 156 T + 3562 T^{2} - 769392 T^{3} + 176051379 T^{4} - 769392 p^{2} T^{5} + 3562 p^{4} T^{6} - 156 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
97 \( ( 1 - 68 T + 3046 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
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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

−4.82319831115590960992665917584, −4.65081628809979034074334597713, −4.27357021044391494376290438689, −4.19982011388231300619091439003, −4.03418178907144871840195362107, −3.96128297498935552097060277628, −3.91594738059435677792295308182, −3.48887820992028248865261312191, −3.31939145905828591441183622329, −3.26528809515673103005807961746, −3.20444699263655240049266260714, −3.17179893582484471589361714285, −3.11331638665019045023412050970, −2.65145611804196579478027461112, −2.46774194023061024667783146000, −2.26493459326438016052876621316, −2.20546257870998399001972504342, −2.09122054490692145395137687905, −1.98366001459634990492369042085, −1.58105303880056688812369512005, −1.26464481815968631257665005116, −1.20217601216040448057079297015, −0.959894915781583229640164619873, −0.72870140849621422416125330254, −0.64136042506000896168684520372, 0.64136042506000896168684520372, 0.72870140849621422416125330254, 0.959894915781583229640164619873, 1.20217601216040448057079297015, 1.26464481815968631257665005116, 1.58105303880056688812369512005, 1.98366001459634990492369042085, 2.09122054490692145395137687905, 2.20546257870998399001972504342, 2.26493459326438016052876621316, 2.46774194023061024667783146000, 2.65145611804196579478027461112, 3.11331638665019045023412050970, 3.17179893582484471589361714285, 3.20444699263655240049266260714, 3.26528809515673103005807961746, 3.31939145905828591441183622329, 3.48887820992028248865261312191, 3.91594738059435677792295308182, 3.96128297498935552097060277628, 4.03418178907144871840195362107, 4.19982011388231300619091439003, 4.27357021044391494376290438689, 4.65081628809979034074334597713, 4.82319831115590960992665917584

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

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