Properties

Degree 40
Conductor $ 2^{40} \cdot 3^{60} $
Sign $1$
Motivic weight 5
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 10·4-s − 116·13-s − 992·16-s + 2.36e4·25-s − 6.45e3·37-s + 1.45e5·49-s − 1.16e3·52-s + 8.46e4·61-s − 1.13e4·64-s + 8.54e4·73-s − 2.19e5·97-s + 2.36e5·100-s + 1.85e5·109-s − 1.64e6·121-s + 127-s + 131-s + 137-s + 139-s − 6.45e4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 3.15e6·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 5/16·4-s − 0.190·13-s − 0.968·16-s + 7.56·25-s − 0.774·37-s + 8.67·49-s − 0.0594·52-s + 2.91·61-s − 0.345·64-s + 1.87·73-s − 2.37·97-s + 2.36·100-s + 1.49·109-s − 10.2·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 0.242·148-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 − 8.50·169-s + 2.54e−6·173-s + 2.33e−6·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{60}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{60}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(40\)
\( N \)  =  \(2^{40} \cdot 3^{60}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{108} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((40,\ 2^{40} \cdot 3^{60} ,\ ( \ : [5/2]^{20} ),\ 1 )\)
\(L(3)\)  \(\approx\)  \(55.7445\)
\(L(\frac12)\)  \(\approx\)  \(55.7445\)
\(L(\frac{7}{2})\)   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 40. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 39.
$p$$F_p(T)$
bad2 \( 1 - 5 p T^{2} + 273 p^{2} T^{4} - 297 p^{5} T^{6} - 5025 p^{8} T^{8} - 93 p^{13} T^{10} - 5025 p^{18} T^{12} - 297 p^{25} T^{14} + 273 p^{32} T^{16} - 5 p^{41} T^{18} + p^{50} T^{20} \)
3 \( 1 \)
good5 \( ( 1 - 11818 T^{2} + 71630853 T^{4} - 313823973048 T^{6} + 1185932779192818 T^{8} - 3986228637264265212 T^{10} + 1185932779192818 p^{10} T^{12} - 313823973048 p^{20} T^{14} + 71630853 p^{30} T^{16} - 11818 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
7 \( ( 1 - 10417 p T^{2} + 417769113 p T^{4} - 81340986295962 T^{6} + 1774418094261308301 T^{8} - \)\(32\!\cdots\!57\)\( T^{10} + 1774418094261308301 p^{10} T^{12} - 81340986295962 p^{20} T^{14} + 417769113 p^{31} T^{16} - 10417 p^{41} T^{18} + p^{50} T^{20} )^{2} \)
11 \( ( 1 + 822230 T^{2} + 360495679845 T^{4} + 108431998645925448 T^{6} + \)\(24\!\cdots\!70\)\( T^{8} + \)\(44\!\cdots\!92\)\( T^{10} + \)\(24\!\cdots\!70\)\( p^{10} T^{12} + 108431998645925448 p^{20} T^{14} + 360495679845 p^{30} T^{16} + 822230 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
13 \( ( 1 + 29 T + 60855 p T^{2} + 39777006 T^{3} + 380980388709 T^{4} + 35518566649227 T^{5} + 380980388709 p^{5} T^{6} + 39777006 p^{10} T^{7} + 60855 p^{16} T^{8} + 29 p^{20} T^{9} + p^{25} T^{10} )^{4} \)
17 \( ( 1 - 6202690 T^{2} + 18689060069517 T^{4} - 34705007722821565080 T^{6} + \)\(27\!\cdots\!06\)\( p T^{8} - \)\(60\!\cdots\!16\)\( T^{10} + \)\(27\!\cdots\!06\)\( p^{11} T^{12} - 34705007722821565080 p^{20} T^{14} + 18689060069517 p^{30} T^{16} - 6202690 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
19 \( ( 1 - 13782439 T^{2} + 98755621335207 T^{4} - \)\(47\!\cdots\!14\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} - \)\(48\!\cdots\!01\)\( T^{10} + \)\(17\!\cdots\!01\)\( p^{10} T^{12} - \)\(47\!\cdots\!14\)\( p^{20} T^{14} + 98755621335207 p^{30} T^{16} - 13782439 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
23 \( ( 1 + 42721646 T^{2} + 918206452664445 T^{4} + \)\(12\!\cdots\!28\)\( T^{6} + \)\(12\!\cdots\!34\)\( T^{8} + \)\(95\!\cdots\!68\)\( T^{10} + \)\(12\!\cdots\!34\)\( p^{10} T^{12} + \)\(12\!\cdots\!28\)\( p^{20} T^{14} + 918206452664445 p^{30} T^{16} + 42721646 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
29 \( ( 1 - 142578466 T^{2} + 336784537390521 p T^{4} - \)\(42\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!22\)\( T^{8} - \)\(31\!\cdots\!16\)\( T^{10} + \)\(13\!\cdots\!22\)\( p^{10} T^{12} - \)\(42\!\cdots\!00\)\( p^{20} T^{14} + 336784537390521 p^{31} T^{16} - 142578466 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
31 \( ( 1 - 53059450 T^{2} + 1830387109253997 T^{4} - \)\(54\!\cdots\!08\)\( T^{6} + \)\(17\!\cdots\!98\)\( T^{8} - \)\(63\!\cdots\!76\)\( T^{10} + \)\(17\!\cdots\!98\)\( p^{10} T^{12} - \)\(54\!\cdots\!08\)\( p^{20} T^{14} + 1830387109253997 p^{30} T^{16} - 53059450 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
37 \( ( 1 + 1613 T + 123026979 T^{2} + 343711884654 T^{3} + 5803261980890325 T^{4} + 31670451235762477275 T^{5} + 5803261980890325 p^{5} T^{6} + 343711884654 p^{10} T^{7} + 123026979 p^{15} T^{8} + 1613 p^{20} T^{9} + p^{25} T^{10} )^{4} \)
41 \( ( 1 - 204662074 T^{2} + 6421654936070589 T^{4} + \)\(14\!\cdots\!28\)\( T^{6} - \)\(52\!\cdots\!06\)\( T^{8} - \)\(96\!\cdots\!76\)\( T^{10} - \)\(52\!\cdots\!06\)\( p^{10} T^{12} + \)\(14\!\cdots\!28\)\( p^{20} T^{14} + 6421654936070589 p^{30} T^{16} - 204662074 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
43 \( ( 1 - 1038109906 T^{2} + 527969585590823397 T^{4} - \)\(17\!\cdots\!40\)\( T^{6} + \)\(39\!\cdots\!02\)\( T^{8} - \)\(67\!\cdots\!12\)\( T^{10} + \)\(39\!\cdots\!02\)\( p^{10} T^{12} - \)\(17\!\cdots\!40\)\( p^{20} T^{14} + 527969585590823397 p^{30} T^{16} - 1038109906 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
47 \( ( 1 + 920931614 T^{2} + 499070896097327373 T^{4} + \)\(18\!\cdots\!24\)\( T^{6} + \)\(53\!\cdots\!58\)\( T^{8} + \)\(12\!\cdots\!24\)\( T^{10} + \)\(53\!\cdots\!58\)\( p^{10} T^{12} + \)\(18\!\cdots\!24\)\( p^{20} T^{14} + 499070896097327373 p^{30} T^{16} + 920931614 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
53 \( ( 1 - 1301597650 T^{2} + 1197727639041193893 T^{4} - \)\(82\!\cdots\!04\)\( T^{6} + \)\(45\!\cdots\!18\)\( T^{8} - \)\(21\!\cdots\!32\)\( T^{10} + \)\(45\!\cdots\!18\)\( p^{10} T^{12} - \)\(82\!\cdots\!04\)\( p^{20} T^{14} + 1197727639041193893 p^{30} T^{16} - 1301597650 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
59 \( ( 1 + 1806503990 T^{2} + 2218805640623321733 T^{4} + \)\(21\!\cdots\!92\)\( T^{6} + \)\(17\!\cdots\!70\)\( T^{8} + \)\(12\!\cdots\!12\)\( T^{10} + \)\(17\!\cdots\!70\)\( p^{10} T^{12} + \)\(21\!\cdots\!92\)\( p^{20} T^{14} + 2218805640623321733 p^{30} T^{16} + 1806503990 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
61 \( ( 1 - 21151 T + 2606330283 T^{2} - 19051394797770 T^{3} + 2604033901923348357 T^{4} - \)\(93\!\cdots\!13\)\( T^{5} + 2604033901923348357 p^{5} T^{6} - 19051394797770 p^{10} T^{7} + 2606330283 p^{15} T^{8} - 21151 p^{20} T^{9} + p^{25} T^{10} )^{4} \)
67 \( ( 1 - 10598201455 T^{2} + 53515912648993384359 T^{4} - \)\(16\!\cdots\!10\)\( T^{6} + \)\(36\!\cdots\!89\)\( T^{8} - \)\(58\!\cdots\!01\)\( T^{10} + \)\(36\!\cdots\!89\)\( p^{10} T^{12} - \)\(16\!\cdots\!10\)\( p^{20} T^{14} + 53515912648993384359 p^{30} T^{16} - 10598201455 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
71 \( ( 1 + 6724308230 T^{2} + 20388189580420161501 T^{4} + \)\(45\!\cdots\!64\)\( T^{6} + \)\(10\!\cdots\!26\)\( T^{8} + \)\(22\!\cdots\!68\)\( T^{10} + \)\(10\!\cdots\!26\)\( p^{10} T^{12} + \)\(45\!\cdots\!64\)\( p^{20} T^{14} + 20388189580420161501 p^{30} T^{16} + 6724308230 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
73 \( ( 1 - 21355 T + 2385405975 T^{2} - 91039324043106 T^{3} + 7775807385428270205 T^{4} - \)\(26\!\cdots\!29\)\( T^{5} + 7775807385428270205 p^{5} T^{6} - 91039324043106 p^{10} T^{7} + 2385405975 p^{15} T^{8} - 21355 p^{20} T^{9} + p^{25} T^{10} )^{4} \)
79 \( ( 1 - 13863408967 T^{2} + 96370465581362357535 T^{4} - \)\(46\!\cdots\!74\)\( T^{6} + \)\(17\!\cdots\!65\)\( T^{8} - \)\(59\!\cdots\!89\)\( T^{10} + \)\(17\!\cdots\!65\)\( p^{10} T^{12} - \)\(46\!\cdots\!74\)\( p^{20} T^{14} + 96370465581362357535 p^{30} T^{16} - 13863408967 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
83 \( ( 1 + 15102812222 T^{2} + \)\(16\!\cdots\!49\)\( T^{4} + \)\(11\!\cdots\!68\)\( T^{6} + \)\(64\!\cdots\!90\)\( T^{8} + \)\(28\!\cdots\!80\)\( T^{10} + \)\(64\!\cdots\!90\)\( p^{10} T^{12} + \)\(11\!\cdots\!68\)\( p^{20} T^{14} + \)\(16\!\cdots\!49\)\( p^{30} T^{16} + 15102812222 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
89 \( ( 1 - 46714027858 T^{2} + \)\(10\!\cdots\!53\)\( T^{4} - \)\(13\!\cdots\!72\)\( T^{6} + \)\(12\!\cdots\!82\)\( T^{8} - \)\(83\!\cdots\!68\)\( T^{10} + \)\(12\!\cdots\!82\)\( p^{10} T^{12} - \)\(13\!\cdots\!72\)\( p^{20} T^{14} + \)\(10\!\cdots\!53\)\( p^{30} T^{16} - 46714027858 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
97 \( ( 1 + 54977 T + 25395595455 T^{2} + 1194992001523350 T^{3} + \)\(29\!\cdots\!17\)\( T^{4} + \)\(12\!\cdots\!15\)\( T^{5} + \)\(29\!\cdots\!17\)\( p^{5} T^{6} + 1194992001523350 p^{10} T^{7} + 25395595455 p^{15} T^{8} + 54977 p^{20} T^{9} + p^{25} T^{10} )^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.48737282816977234546048265287, −2.46088327117755660438746606687, −2.46080167187826674060286638360, −2.23234002352648346479458614014, −2.18050458601373869859652385225, −2.15625433743368561657857688147, −2.02801307237571778973182389555, −1.83686416694363202455111489179, −1.60304599912431397563386432224, −1.54070345266395264481326926896, −1.49213774125702166260525825581, −1.37995032228404282001179454752, −1.33864223299079614932844075677, −1.15818913962509286636950076318, −1.08120847582640209705496781823, −1.05500315223204899615386919366, −0.919836542125674704351169530544, −0.904548895537723653401249760891, −0.68096574735287298065108836409, −0.59423156121907396299609677975, −0.55798214124665453491109252392, −0.45133599527974654307545985947, −0.42484036942573626694974850338, −0.19898474730670339305442047665, −0.13036777292092968409116174284, 0.13036777292092968409116174284, 0.19898474730670339305442047665, 0.42484036942573626694974850338, 0.45133599527974654307545985947, 0.55798214124665453491109252392, 0.59423156121907396299609677975, 0.68096574735287298065108836409, 0.904548895537723653401249760891, 0.919836542125674704351169530544, 1.05500315223204899615386919366, 1.08120847582640209705496781823, 1.15818913962509286636950076318, 1.33864223299079614932844075677, 1.37995032228404282001179454752, 1.49213774125702166260525825581, 1.54070345266395264481326926896, 1.60304599912431397563386432224, 1.83686416694363202455111489179, 2.02801307237571778973182389555, 2.15625433743368561657857688147, 2.18050458601373869859652385225, 2.23234002352648346479458614014, 2.46080167187826674060286638360, 2.46088327117755660438746606687, 2.48737282816977234546048265287

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

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