# Properties

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

# Origins of factors

## 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.