LMFDB - L-function 24-275e12-1.1-c5e12-0-0

Dirichlet series

L(s) = 1

+ 77·4^-s + 626·9^-s + 1.45e3·11^-s + 4.46e3·16^-s − 1.22e3·19^-s − 2.19e4·29^-s + 1.77e4·31^-s + 4.82e4·36^-s + 320·41^-s + 1.11e5·44^-s + 5.23e4·49^-s − 2.41e5·59^-s − 1.84e4·61^-s + 1.61e5·64^-s + 3.27e5·71^-s − 9.39e4·76^-s + 7.34e4·79^-s + 1.95e5·81^-s − 5.09e5·89^-s + 9.08e5·99^-s − 2.15e5·101^-s + 4.11e5·109^-s − 1.68e6·116^-s + 1.14e6·121^-s + 1.36e6·124^-s + 127^-s + 131^-s + ⋯

L(s) = 1

+ 2.40·4^-s + 2.57·9^-s + 3.61·11^-s + 4.35·16^-s − 0.775·19^-s − 4.84·29^-s + 3.31·31^-s + 6.19·36^-s + 0.0297·41^-s + 8.70·44^-s + 3.11·49^-s − 9.01·59^-s − 0.635·61^-s + 4.93·64^-s + 7.71·71^-s − 1.86·76^-s + 1.32·79^-s + 3.31·81^-s − 6.82·89^-s + 9.32·99^-s − 2.09·101^-s + 3.31·109^-s − 11.6·116^-s + 7.09·121^-s + 7.97·124^-s + 5.50e−6·127^-s + 5.09e−6·131^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$24$
Conductor:	$5^{24} \cdot 11^{12}$
Sign:	$1$
Analytic conductor:	$5.41899\times 10^{19}$
Root analytic conductor:	$6.64120$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(24,\ 5^{24} \cdot 11^{12} ,\ ( \ : [5/2]^{12} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$32.40152215$
$L(\frac12)$	$\approx$	$32.40152215$
$L(\frac{7}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	5	$ 1 $
	11	$ ( 1 - p^{2} T )^{12} $
good	2	$ 1 - 77 T^{2} + 1467 T^{4} + 68965 T^{6} - 735693 p^{2} T^{8} - 1769085 p^{4} T^{10} + 59246297 p^{6} T^{12} - 1769085 p^{14} T^{14} - 735693 p^{22} T^{16} + 68965 p^{30} T^{18} + 1467 p^{40} T^{20} - 77 p^{50} T^{22} + p^{60} T^{24} $
	3	$ 1 - 626 T^{2} + 195943 T^{4} - 2742370 p^{2} T^{6} + 71354459 p^{4} T^{8} - 4596790220 p^{6} T^{10} + 21464665546 p^{10} T^{12} - 4596790220 p^{16} T^{14} + 71354459 p^{24} T^{16} - 2742370 p^{32} T^{18} + 195943 p^{40} T^{20} - 626 p^{50} T^{22} + p^{60} T^{24} $
	7	$ 1 - 52330 T^{2} + 2304428159 T^{4} - 62787078790250 T^{6} + 1575432931160812155 T^{8} - $$42\!\cdots\!00$$ p T^{10} + $$56\!\cdots\!70$$ T^{12} - $$42\!\cdots\!00$$ p^{11} T^{14} + 1575432931160812155 p^{20} T^{16} - 62787078790250 p^{30} T^{18} + 2304428159 p^{40} T^{20} - 52330 p^{50} T^{22} + p^{60} T^{24} $
	13	$ 1 - 2502004 T^{2} + 247919930842 p T^{4} - 2788904116613606340 T^{6} + $$18\!\cdots\!27$$ T^{8} - $$91\!\cdots\!60$$ T^{10} + $$37\!\cdots\!52$$ T^{12} - $$91\!\cdots\!60$$ p^{10} T^{14} + $$18\!\cdots\!27$$ p^{20} T^{16} - 2788904116613606340 p^{30} T^{18} + 247919930842 p^{41} T^{20} - 2502004 p^{50} T^{22} + p^{60} T^{24} $
	17	$ 1 - 3106170 T^{2} + 7609381403807 T^{4} - 17726097284463026170 T^{6} + $$33\!\cdots\!43$$ T^{8} - $$56\!\cdots\!40$$ T^{10} + $$86\!\cdots\!98$$ T^{12} - $$56\!\cdots\!40$$ p^{10} T^{14} + $$33\!\cdots\!43$$ p^{20} T^{16} - 17726097284463026170 p^{30} T^{18} + 7609381403807 p^{40} T^{20} - 3106170 p^{50} T^{22} + p^{60} T^{24} $
	19	$ ( 1 + 610 T + 3190579 T^{2} + 6190567550 T^{3} + 12814117289875 T^{4} + 16505267615365300 T^{5} + 41329816265517558290 T^{6} + 16505267615365300 p^{5} T^{7} + 12814117289875 p^{10} T^{8} + 6190567550 p^{15} T^{9} + 3190579 p^{20} T^{10} + 610 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	23	$ 1 - 42001356 T^{2} + 796007446589378 T^{4} - $$88\!\cdots\!20$$ T^{6} + $$61\!\cdots\!79$$ T^{8} - $$30\!\cdots\!60$$ T^{10} + $$14\!\cdots\!84$$ T^{12} - $$30\!\cdots\!60$$ p^{10} T^{14} + $$61\!\cdots\!79$$ p^{20} T^{16} - $$88\!\cdots\!20$$ p^{30} T^{18} + 796007446589378 p^{40} T^{20} - 42001356 p^{50} T^{22} + p^{60} T^{24} $
	29	$ ( 1 + 10964 T + 122520281 T^{2} + 815259092220 T^{3} + 5431302911221787 T^{4} + 26728592086363454720 T^{5} + $$13\!\cdots\!62$$ T^{6} + 26728592086363454720 p^{5} T^{7} + 5431302911221787 p^{10} T^{8} + 815259092220 p^{15} T^{9} + 122520281 p^{20} T^{10} + 10964 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	31	$ ( 1 - 8868 T + 23486765 T^{2} + 174911358708 T^{3} - 765101731238653 T^{4} - 7337521350847229736 T^{5} + $$76\!\cdots\!66$$ T^{6} - 7337521350847229736 p^{5} T^{7} - 765101731238653 p^{10} T^{8} + 174911358708 p^{15} T^{9} + 23486765 p^{20} T^{10} - 8868 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	37	$ 1 - 340568146 T^{2} + 60103698765473431 T^{4} - $$69\!\cdots\!10$$ T^{6} + $$58\!\cdots\!47$$ T^{8} - $$41\!\cdots\!40$$ T^{10} + $$27\!\cdots\!42$$ T^{12} - $$41\!\cdots\!40$$ p^{10} T^{14} + $$58\!\cdots\!47$$ p^{20} T^{16} - $$69\!\cdots\!10$$ p^{30} T^{18} + 60103698765473431 p^{40} T^{20} - 340568146 p^{50} T^{22} + p^{60} T^{24} $
	41	$ ( 1 - 160 T + 462530202 T^{2} - 777925073248 T^{3} + 2599242825336423 p T^{4} - $$18\!\cdots\!08$$ T^{5} + $$15\!\cdots\!40$$ T^{6} - $$18\!\cdots\!08$$ p^{5} T^{7} + 2599242825336423 p^{11} T^{8} - 777925073248 p^{15} T^{9} + 462530202 p^{20} T^{10} - 160 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	43	$ 1 - 1132814260 T^{2} + 560368076182513394 T^{4} - $$15\!\cdots\!00$$ T^{6} + $$26\!\cdots\!15$$ T^{8} - $$28\!\cdots\!00$$ T^{10} + $$29\!\cdots\!80$$ T^{12} - $$28\!\cdots\!00$$ p^{10} T^{14} + $$26\!\cdots\!15$$ p^{20} T^{16} - $$15\!\cdots\!00$$ p^{30} T^{18} + 560368076182513394 p^{40} T^{20} - 1132814260 p^{50} T^{22} + p^{60} T^{24} $
	47	$ 1 - 26382836 p T^{2} + 760499985411312290 T^{4} - $$31\!\cdots\!60$$ T^{6} + $$10\!\cdots\!71$$ T^{8} - $$27\!\cdots\!60$$ T^{10} + $$66\!\cdots\!76$$ T^{12} - $$27\!\cdots\!60$$ p^{10} T^{14} + $$10\!\cdots\!71$$ p^{20} T^{16} - $$31\!\cdots\!60$$ p^{30} T^{18} + 760499985411312290 p^{40} T^{20} - 26382836 p^{51} T^{22} + p^{60} T^{24} $
	53	$ 1 - 3045659586 T^{2} + 84156867637537675 p T^{4} - $$42\!\cdots\!90$$ T^{6} + $$29\!\cdots\!51$$ T^{8} - $$16\!\cdots\!00$$ T^{10} + $$74\!\cdots\!46$$ T^{12} - $$16\!\cdots\!00$$ p^{10} T^{14} + $$29\!\cdots\!51$$ p^{20} T^{16} - $$42\!\cdots\!90$$ p^{30} T^{18} + 84156867637537675 p^{41} T^{20} - 3045659586 p^{50} T^{22} + p^{60} T^{24} $
	59	$ ( 1 + 120532 T + 9412440198 T^{2} + 8753288078820 p T^{3} + 22598394277814176839 T^{4} + $$79\!\cdots\!00$$ T^{5} + $$23\!\cdots\!24$$ T^{6} + $$79\!\cdots\!00$$ p^{5} T^{7} + 22598394277814176839 p^{10} T^{8} + 8753288078820 p^{16} T^{9} + 9412440198 p^{20} T^{10} + 120532 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	61	$ ( 1 + 9228 T + 3228518633 T^{2} + 2450917415284 T^{3} + 4521434490158831931 T^{4} - $$24\!\cdots\!44$$ T^{5} + $$42\!\cdots\!34$$ T^{6} - $$24\!\cdots\!44$$ p^{5} T^{7} + 4521434490158831931 p^{10} T^{8} + 2450917415284 p^{15} T^{9} + 3228518633 p^{20} T^{10} + 9228 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	67	$ 1 - 7922414052 T^{2} + 31435558402452067602 T^{4} - $$86\!\cdots\!80$$ T^{6} + $$18\!\cdots\!83$$ T^{8} - $$33\!\cdots\!00$$ T^{10} + $$49\!\cdots\!28$$ T^{12} - $$33\!\cdots\!00$$ p^{10} T^{14} + $$18\!\cdots\!83$$ p^{20} T^{16} - $$86\!\cdots\!80$$ p^{30} T^{18} + 31435558402452067602 p^{40} T^{20} - 7922414052 p^{50} T^{22} + p^{60} T^{24} $
	71	$ ( 1 - 163948 T + 20114910917 T^{2} - 1708534305484068 T^{3} + $$11\!\cdots\!63$$ T^{4} - $$65\!\cdots\!48$$ T^{5} + $$30\!\cdots\!66$$ T^{6} - $$65\!\cdots\!48$$ p^{5} T^{7} + $$11\!\cdots\!63$$ p^{10} T^{8} - 1708534305484068 p^{15} T^{9} + 20114910917 p^{20} T^{10} - 163948 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	73	$ 1 - 16115400964 T^{2} + $$12\!\cdots\!66$$ T^{4} - $$61\!\cdots\!00$$ T^{6} + $$21\!\cdots\!07$$ T^{8} - $$60\!\cdots\!80$$ T^{10} + $$13\!\cdots\!52$$ T^{12} - $$60\!\cdots\!80$$ p^{10} T^{14} + $$21\!\cdots\!07$$ p^{20} T^{16} - $$61\!\cdots\!00$$ p^{30} T^{18} + $$12\!\cdots\!66$$ p^{40} T^{20} - 16115400964 p^{50} T^{22} + p^{60} T^{24} $
	79	$ ( 1 - 36712 T + 8265852230 T^{2} - 510532153825080 T^{3} + 43291427241491002111 T^{4} - $$23\!\cdots\!00$$ T^{5} + $$17\!\cdots\!16$$ T^{6} - $$23\!\cdots\!00$$ p^{5} T^{7} + 43291427241491002111 p^{10} T^{8} - 510532153825080 p^{15} T^{9} + 8265852230 p^{20} T^{10} - 36712 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	83	$ 1 - 35266245740 T^{2} + $$56\!\cdots\!54$$ T^{4} - $$55\!\cdots\!00$$ T^{6} + $$37\!\cdots\!75$$ T^{8} - $$19\!\cdots\!00$$ T^{10} + $$84\!\cdots\!40$$ T^{12} - $$19\!\cdots\!00$$ p^{10} T^{14} + $$37\!\cdots\!75$$ p^{20} T^{16} - $$55\!\cdots\!00$$ p^{30} T^{18} + $$56\!\cdots\!54$$ p^{40} T^{20} - 35266245740 p^{50} T^{22} + p^{60} T^{24} $
	89	$ ( 1 + 254946 T + 50054578375 T^{2} + 7054717254835170 T^{3} + $$81\!\cdots\!91$$ T^{4} + $$78\!\cdots\!60$$ T^{5} + $$63\!\cdots\!66$$ T^{6} + $$78\!\cdots\!60$$ p^{5} T^{7} + $$81\!\cdots\!91$$ p^{10} T^{8} + 7054717254835170 p^{15} T^{9} + 50054578375 p^{20} T^{10} + 254946 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	97	$ 1 - 88248569836 T^{2} + $$36\!\cdots\!26$$ T^{4} - $$94\!\cdots\!00$$ T^{6} + $$17\!\cdots\!07$$ T^{8} - $$22\!\cdots\!20$$ T^{10} + $$22\!\cdots\!32$$ T^{12} - $$22\!\cdots\!20$$ p^{10} T^{14} + $$17\!\cdots\!07$$ p^{20} T^{16} - $$94\!\cdots\!00$$ p^{30} T^{18} + $$36\!\cdots\!26$$ p^{40} T^{20} - 88248569836 p^{50} T^{22} + p^{60} T^{24} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−3.00943546094049797410031210521, −2.89053605389714983146751361161, −2.82087264961347484575799900368, −2.78942061022368538595288464086, −2.78793500725743495945205102634, −2.46815197469097711727930561490, −2.29055570921911243024239671679, −2.28512277896080033442998422123, −1.96000521789776172245130001345, −1.94265943884237216499490725978, −1.84368343499484093925322914639, −1.72267244196284062309727157595, −1.70746476598355787844471233284, −1.63304860749762783284008692106, −1.47757537467042051641448605216, −1.27695965252926868231924288914, −1.27082984619456796461629455057, −1.25586940125013547231426739294, −0.842264891715174175222154475171, −0.831636359775579765073359536972, −0.812620383026499943380328858021, −0.61380483320727227938126392892, −0.47485203946386640160585549925, −0.26550490016192203829935656641, −0.079393949221094661740183431701, 0.079393949221094661740183431701, 0.26550490016192203829935656641, 0.47485203946386640160585549925, 0.61380483320727227938126392892, 0.812620383026499943380328858021, 0.831636359775579765073359536972, 0.842264891715174175222154475171, 1.25586940125013547231426739294, 1.27082984619456796461629455057, 1.27695965252926868231924288914, 1.47757537467042051641448605216, 1.63304860749762783284008692106, 1.70746476598355787844471233284, 1.72267244196284062309727157595, 1.84368343499484093925322914639, 1.94265943884237216499490725978, 1.96000521789776172245130001345, 2.28512277896080033442998422123, 2.29055570921911243024239671679, 2.46815197469097711727930561490, 2.78793500725743495945205102634, 2.78942061022368538595288464086, 2.82087264961347484575799900368, 2.89053605389714983146751361161, 3.00943546094049797410031210521

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 $\F_{q}$
Belyi maps

\(L(3)\)	\(\approx\)	\(32.40152215\)
\(L(\frac12)\)	\(\approx\)	\(32.40152215\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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