Properties

Degree 32
Conductor $ 2^{32} \cdot 3^{48} $
Sign $1$
Motivic weight 5
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 47·4-s + 896·13-s + 1.14e3·16-s + 2.99e4·25-s − 7.10e4·37-s + 1.09e5·49-s + 4.21e4·52-s + 7.78e4·61-s + 3.42e4·64-s − 3.80e4·73-s − 976·97-s + 1.40e6·100-s + 1.33e5·109-s − 1.48e6·121-s + 127-s + 131-s + 137-s + 139-s − 3.33e6·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.71e6·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 1.46·4-s + 1.47·13-s + 1.12·16-s + 9.58·25-s − 8.52·37-s + 6.53·49-s + 2.15·52-s + 2.68·61-s + 1.04·64-s − 0.835·73-s − 0.0105·97-s + 14.0·100-s + 1.07·109-s − 9.20·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 12.5·148-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 4.61·169-s + 2.54e−6·173-s + 2.33e−6·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(2^{32} \cdot 3^{48}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{108} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((32,\ 2^{32} \cdot 3^{48} ,\ ( \ : [5/2]^{16} ),\ 1 )\)
\(L(3)\)  \(\approx\)  \(142.979\)
\(L(\frac12)\)  \(\approx\)  \(142.979\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 32. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 \( 1 - 47 T^{2} + 265 p^{2} T^{4} - 235 p^{7} T^{6} + 745 p^{10} T^{8} - 235 p^{17} T^{10} + 265 p^{22} T^{12} - 47 p^{30} T^{14} + p^{40} T^{16} \)
3 \( 1 \)
good5 \( ( 1 - 14972 T^{2} + 119342578 T^{4} - 617133972224 T^{6} + 90872805142579 p^{2} T^{8} - 617133972224 p^{10} T^{10} + 119342578 p^{20} T^{12} - 14972 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
7 \( ( 1 - 54884 T^{2} + 1251149938 T^{4} - 15826547698400 T^{6} + 189674874534565723 T^{8} - 15826547698400 p^{10} T^{10} + 1251149938 p^{20} T^{12} - 54884 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
11 \( ( 1 + 741148 T^{2} + 299964078250 T^{4} + 79656648391789648 T^{6} + \)\(15\!\cdots\!99\)\( T^{8} + 79656648391789648 p^{10} T^{10} + 299964078250 p^{20} T^{12} + 741148 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
13 \( ( 1 - 224 T + 554236 T^{2} + 77263648 T^{3} + 200668790230 T^{4} + 77263648 p^{5} T^{5} + 554236 p^{10} T^{6} - 224 p^{15} T^{7} + p^{20} T^{8} )^{4} \)
17 \( ( 1 - 4946936 T^{2} + 15515554849948 T^{4} - 34385249721915417800 T^{6} + \)\(55\!\cdots\!90\)\( T^{8} - 34385249721915417800 p^{10} T^{10} + 15515554849948 p^{20} T^{12} - 4946936 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
19 \( ( 1 - 9713816 T^{2} + 55349997016060 T^{4} - \)\(21\!\cdots\!72\)\( T^{6} + \)\(61\!\cdots\!02\)\( T^{8} - \)\(21\!\cdots\!72\)\( p^{10} T^{10} + 55349997016060 p^{20} T^{12} - 9713816 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
23 \( ( 1 + 29994040 T^{2} + 487786705761244 T^{4} + \)\(51\!\cdots\!44\)\( T^{6} + \)\(39\!\cdots\!14\)\( T^{8} + \)\(51\!\cdots\!44\)\( p^{10} T^{10} + 487786705761244 p^{20} T^{12} + 29994040 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
29 \( ( 1 - 26575640 T^{2} + 859790443267708 T^{4} - \)\(25\!\cdots\!88\)\( T^{6} + \)\(44\!\cdots\!26\)\( T^{8} - \)\(25\!\cdots\!88\)\( p^{10} T^{10} + 859790443267708 p^{20} T^{12} - 26575640 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
31 \( ( 1 - 160216820 T^{2} + 12412163878108546 T^{4} - \)\(60\!\cdots\!08\)\( T^{6} + \)\(20\!\cdots\!87\)\( T^{8} - \)\(60\!\cdots\!08\)\( p^{10} T^{10} + 12412163878108546 p^{20} T^{12} - 160216820 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
37 \( ( 1 + 17752 T + 254119924 T^{2} + 2137611581224 T^{3} + 19827261455867542 T^{4} + 2137611581224 p^{5} T^{5} + 254119924 p^{10} T^{6} + 17752 p^{15} T^{7} + p^{20} T^{8} )^{4} \)
41 \( ( 1 - 667999304 T^{2} + 215857510496501980 T^{4} - \)\(43\!\cdots\!84\)\( T^{6} + \)\(60\!\cdots\!14\)\( T^{8} - \)\(43\!\cdots\!84\)\( p^{10} T^{10} + 215857510496501980 p^{20} T^{12} - 667999304 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
43 \( ( 1 - 271892936 T^{2} + 81822516500494204 T^{4} - \)\(14\!\cdots\!72\)\( T^{6} + \)\(26\!\cdots\!02\)\( T^{8} - \)\(14\!\cdots\!72\)\( p^{10} T^{10} + 81822516500494204 p^{20} T^{12} - 271892936 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
47 \( ( 1 + 1268760328 T^{2} + 775645167851419804 T^{4} + \)\(30\!\cdots\!92\)\( T^{6} + \)\(81\!\cdots\!58\)\( T^{8} + \)\(30\!\cdots\!92\)\( p^{10} T^{10} + 775645167851419804 p^{20} T^{12} + 1268760328 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
53 \( ( 1 - 1665548972 T^{2} + 1494189343182800962 T^{4} - \)\(95\!\cdots\!20\)\( T^{6} + \)\(46\!\cdots\!75\)\( T^{8} - \)\(95\!\cdots\!20\)\( p^{10} T^{10} + 1494189343182800962 p^{20} T^{12} - 1665548972 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
59 \( ( 1 + 4822767208 T^{2} + 10641078380871180220 T^{4} + \)\(14\!\cdots\!92\)\( T^{6} + \)\(12\!\cdots\!22\)\( T^{8} + \)\(14\!\cdots\!92\)\( p^{10} T^{10} + 10641078380871180220 p^{20} T^{12} + 4822767208 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
61 \( ( 1 - 19472 T + 1261489684 T^{2} - 43949943879536 T^{3} + 833752337244254902 T^{4} - 43949943879536 p^{5} T^{5} + 1261489684 p^{10} T^{6} - 19472 p^{15} T^{7} + p^{20} T^{8} )^{4} \)
67 \( ( 1 - 1447449416 T^{2} + 2915553445416739516 T^{4} - \)\(52\!\cdots\!68\)\( T^{6} + \)\(74\!\cdots\!58\)\( T^{8} - \)\(52\!\cdots\!68\)\( p^{10} T^{10} + 2915553445416739516 p^{20} T^{12} - 1447449416 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
71 \( ( 1 + 9830208232 T^{2} + 46886702724166517020 T^{4} + \)\(14\!\cdots\!88\)\( T^{6} + \)\(30\!\cdots\!22\)\( T^{8} + \)\(14\!\cdots\!88\)\( p^{10} T^{10} + 46886702724166517020 p^{20} T^{12} + 9830208232 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
73 \( ( 1 + 9508 T + 3097043794 T^{2} - 21796113866240 T^{3} + 7953703459188333211 T^{4} - 21796113866240 p^{5} T^{5} + 3097043794 p^{10} T^{6} + 9508 p^{15} T^{7} + p^{20} T^{8} )^{4} \)
79 \( ( 1 + 99394360 T^{2} + 19778301042529431580 T^{4} - \)\(22\!\cdots\!96\)\( T^{6} + \)\(20\!\cdots\!66\)\( T^{8} - \)\(22\!\cdots\!96\)\( p^{10} T^{10} + 19778301042529431580 p^{20} T^{12} + 99394360 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
83 \( ( 1 + 20559730876 T^{2} + \)\(21\!\cdots\!14\)\( T^{4} + \)\(14\!\cdots\!68\)\( T^{6} + \)\(70\!\cdots\!39\)\( T^{8} + \)\(14\!\cdots\!68\)\( p^{10} T^{10} + \)\(21\!\cdots\!14\)\( p^{20} T^{12} + 20559730876 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
89 \( ( 1 - 30119375768 T^{2} + \)\(43\!\cdots\!12\)\( T^{4} - \)\(41\!\cdots\!60\)\( T^{6} + \)\(27\!\cdots\!18\)\( T^{8} - \)\(41\!\cdots\!60\)\( p^{10} T^{10} + \)\(43\!\cdots\!12\)\( p^{20} T^{12} - 30119375768 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
97 \( ( 1 + 244 T + 6580710202 T^{2} + 1109461517844208 T^{3} - 18256445891874654653 T^{4} + 1109461517844208 p^{5} T^{5} + 6580710202 p^{10} T^{6} + 244 p^{15} T^{7} + p^{20} T^{8} )^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.06559961341032869074544483724, −2.89337951459957598895076648727, −2.86485919256637066185294177443, −2.83680524468104926567799548072, −2.49158167117141460440320834680, −2.35053125945215032546958628824, −2.34409145973516800983621999572, −2.30197582416654672140997743798, −2.18032294409177030409578709982, −2.11534485428649399393689800776, −2.09374092474360616583270230193, −1.50439025118279309023963818542, −1.46311192804971778677582882493, −1.46142461243087437656755537886, −1.40239142103466940700385007682, −1.39859700175376137860115695359, −1.19193493106104924265191653367, −1.04263375538950741189301661293, −0.977159774007818430844266135207, −0.795006178458790688793342862122, −0.76534862495844142980071814979, −0.39815308574753285575646943263, −0.38632454356948183721764351549, −0.32408973754381589247151495447, −0.28724925895830211573003747896, 0.28724925895830211573003747896, 0.32408973754381589247151495447, 0.38632454356948183721764351549, 0.39815308574753285575646943263, 0.76534862495844142980071814979, 0.795006178458790688793342862122, 0.977159774007818430844266135207, 1.04263375538950741189301661293, 1.19193493106104924265191653367, 1.39859700175376137860115695359, 1.40239142103466940700385007682, 1.46142461243087437656755537886, 1.46311192804971778677582882493, 1.50439025118279309023963818542, 2.09374092474360616583270230193, 2.11534485428649399393689800776, 2.18032294409177030409578709982, 2.30197582416654672140997743798, 2.34409145973516800983621999572, 2.35053125945215032546958628824, 2.49158167117141460440320834680, 2.83680524468104926567799548072, 2.86485919256637066185294177443, 2.89337951459957598895076648727, 3.06559961341032869074544483724

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

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