LMFDB - L-function 40-108e20-1.1-c5e20-0-0

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

Degree:	$40$
Conductor:	$2^{40} \cdot 3^{60}$
Sign:	$1$
Analytic conductor:	$5.91134\times 10^{24}$
Root analytic conductor:	$4.16190$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(40,\ 2^{40} \cdot 3^{60} ,\ ( \ : [5/2]^{20} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$55.74452951$
$L(\frac12)$	$\approx$	$55.74452951$
$L(\frac{7}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 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 $
good	5	$ ( 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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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

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.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(3)\)	\(\approx\)	\(55.74452951\)
\(L(\frac12)\)	\(\approx\)	\(55.74452951\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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