LMFDB - L-function 24-338e12-1.1-c5e12-0-0

Dirichlet series

L(s) = 1

− 72·3^-s − 96·4^-s + 1.42e3·9^-s + 6.91e3·12^-s + 5.37e3·16^-s + 1.96e3·17^-s + 1.08e4·23^-s + 2.12e4·25^-s + 1.71e4·27^-s + 6.55e3·29^-s − 1.37e5·36^-s + 3.21e4·43^-s − 3.87e5·48^-s + 1.62e5·49^-s − 1.41e5·51^-s − 4.46e4·53^-s − 1.36e5·61^-s − 2.29e5·64^-s − 1.88e5·68^-s − 7.78e5·69^-s − 1.53e6·75^-s − 1.32e5·79^-s − 9.21e5·81^-s − 4.72e5·87^-s − 1.03e6·92^-s − 2.04e6·100^-s + 2.58e5·101^-s + ⋯

L(s) = 1

− 4.61·3^-s − 3·4^-s + 5.87·9^-s + 13.8·12^-s + 21/4·16^-s + 1.64·17^-s + 4.26·23^-s + 6.80·25^-s + 4.53·27^-s + 1.44·29^-s − 17.6·36^-s + 2.64·43^-s − 24.2·48^-s + 9.66·49^-s − 7.59·51^-s − 2.18·53^-s − 4.69·61^-s − 7·64^-s − 4.93·68^-s − 19.6·69^-s − 31.4·75^-s − 2.38·79^-s − 15.6·81^-s − 6.68·87^-s − 12.7·92^-s − 20.4·100^-s + 2.52·101^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$24$
Conductor:	$2^{12} \cdot 13^{24}$
Sign:	$1$
Analytic conductor:	$6.44061\times 10^{20}$
Root analytic conductor:	$7.36272$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(24,\ 2^{12} \cdot 13^{24} ,\ ( \ : [5/2]^{12} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$0.9538437019$
$L(\frac12)$	$\approx$	$0.9538437019$
$L(\frac{7}{2})$		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 + p^{4} T^{2} )^{6} $
	13	$ 1 $
good	3	$ ( 1 + 4 p^{2} T + 410 p T^{2} + 30944 T^{3} + 78494 p^{2} T^{4} + 1466794 p^{2} T^{5} + 8145559 p^{3} T^{6} + 1466794 p^{7} T^{7} + 78494 p^{12} T^{8} + 30944 p^{15} T^{9} + 410 p^{21} T^{10} + 4 p^{27} T^{11} + p^{30} T^{12} )^{2} $
	5	$ 1 - 21272 T^{2} + 211527624 T^{4} - 1314676986882 T^{6} + 233797705572432 p^{2} T^{8} - 20806687438165628592 T^{10} + $$66\!\cdots\!79$$ T^{12} - 20806687438165628592 p^{10} T^{14} + 233797705572432 p^{22} T^{16} - 1314676986882 p^{30} T^{18} + 211527624 p^{40} T^{20} - 21272 p^{50} T^{22} + p^{60} T^{24} $
	7	$ 1 - 162502 T^{2} + 12562749307 T^{4} - 611612142299164 T^{6} + 2986331094606780247 p T^{8} - $$10\!\cdots\!58$$ p^{2} T^{10} + $$20\!\cdots\!75$$ p^{2} T^{12} - $$10\!\cdots\!58$$ p^{12} T^{14} + 2986331094606780247 p^{21} T^{16} - 611612142299164 p^{30} T^{18} + 12562749307 p^{40} T^{20} - 162502 p^{50} T^{22} + p^{60} T^{24} $
	11	$ 1 - 1187236 T^{2} + 705589763408 T^{4} - 278602603717994762 T^{6} + $$81\!\cdots\!76$$ T^{8} - $$18\!\cdots\!04$$ T^{10} + $$33\!\cdots\!75$$ T^{12} - $$18\!\cdots\!04$$ p^{10} T^{14} + $$81\!\cdots\!76$$ p^{20} T^{16} - 278602603717994762 p^{30} T^{18} + 705589763408 p^{40} T^{20} - 1187236 p^{50} T^{22} + p^{60} T^{24} $
	17	$ ( 1 - 980 T + 5433784 T^{2} - 4928833714 T^{3} + 13758820110808 T^{4} - 11762601649747648 T^{5} + 22893103492509283075 T^{6} - 11762601649747648 p^{5} T^{7} + 13758820110808 p^{10} T^{8} - 4928833714 p^{15} T^{9} + 5433784 p^{20} T^{10} - 980 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	19	$ 1 - 17357500 T^{2} + 154229688109308 T^{4} - $$92\!\cdots\!50$$ T^{6} + $$41\!\cdots\!12$$ T^{8} - $$14\!\cdots\!20$$ T^{10} + $$39\!\cdots\!79$$ T^{12} - $$14\!\cdots\!20$$ p^{10} T^{14} + $$41\!\cdots\!12$$ p^{20} T^{16} - $$92\!\cdots\!50$$ p^{30} T^{18} + 154229688109308 p^{40} T^{20} - 17357500 p^{50} T^{22} + p^{60} T^{24} $
	23	$ ( 1 - 5408 T + 30022239 T^{2} - 99076121168 T^{3} + 350443092650398 T^{4} - 885852301500460448 T^{5} + $$25\!\cdots\!71$$ T^{6} - 885852301500460448 p^{5} T^{7} + 350443092650398 p^{10} T^{8} - 99076121168 p^{15} T^{9} + 30022239 p^{20} T^{10} - 5408 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	29	$ ( 1 - 3278 T + 81260211 T^{2} - 252294509008 T^{3} + 3420929052034245 T^{4} - 9130400891772958686 T^{5} + $$86\!\cdots\!35$$ T^{6} - 9130400891772958686 p^{5} T^{7} + 3420929052034245 p^{10} T^{8} - 252294509008 p^{15} T^{9} + 81260211 p^{20} T^{10} - 3278 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	31	$ 1 - 298934606 T^{2} + 41987933783418427 T^{4} - $$36\!\cdots\!80$$ T^{6} + $$22\!\cdots\!61$$ T^{8} - $$97\!\cdots\!54$$ T^{10} + $$32\!\cdots\!43$$ T^{12} - $$97\!\cdots\!54$$ p^{10} T^{14} + $$22\!\cdots\!61$$ p^{20} T^{16} - $$36\!\cdots\!80$$ p^{30} T^{18} + 41987933783418427 p^{40} T^{20} - 298934606 p^{50} T^{22} + p^{60} T^{24} $
	37	$ 1 - 483817864 T^{2} + 119508967478929816 T^{4} - $$19\!\cdots\!42$$ T^{6} + $$24\!\cdots\!44$$ T^{8} - $$23\!\cdots\!92$$ T^{10} + $$18\!\cdots\!55$$ T^{12} - $$23\!\cdots\!92$$ p^{10} T^{14} + $$24\!\cdots\!44$$ p^{20} T^{16} - $$19\!\cdots\!42$$ p^{30} T^{18} + 119508967478929816 p^{40} T^{20} - 483817864 p^{50} T^{22} + p^{60} T^{24} $
	41	$ 1 - 2242606 T^{2} - 14660280441481091 T^{4} - $$17\!\cdots\!74$$ T^{6} - $$11\!\cdots\!14$$ T^{8} + $$95\!\cdots\!22$$ T^{10} + $$39\!\cdots\!53$$ T^{12} + $$95\!\cdots\!22$$ p^{10} T^{14} - $$11\!\cdots\!14$$ p^{20} T^{16} - $$17\!\cdots\!74$$ p^{30} T^{18} - 14660280441481091 p^{40} T^{20} - 2242606 p^{50} T^{22} + p^{60} T^{24} $
	43	$ ( 1 - 16050 T + 355188653 T^{2} - 6051355408264 T^{3} + 87899237304223639 T^{4} - $$11\!\cdots\!28$$ T^{5} + $$15\!\cdots\!21$$ T^{6} - $$11\!\cdots\!28$$ p^{5} T^{7} + 87899237304223639 p^{10} T^{8} - 6051355408264 p^{15} T^{9} + 355188653 p^{20} T^{10} - 16050 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	47	$ 1 - 1288428420 T^{2} + 858054130838034256 T^{4} - $$38\!\cdots\!62$$ T^{6} + $$13\!\cdots\!00$$ T^{8} - $$37\!\cdots\!56$$ T^{10} + $$90\!\cdots\!75$$ T^{12} - $$37\!\cdots\!56$$ p^{10} T^{14} + $$13\!\cdots\!00$$ p^{20} T^{16} - $$38\!\cdots\!62$$ p^{30} T^{18} + 858054130838034256 p^{40} T^{20} - 1288428420 p^{50} T^{22} + p^{60} T^{24} $
	53	$ ( 1 + 22326 T + 1787665859 T^{2} + 32764270404244 T^{3} + 1541935343094270589 T^{4} + $$23\!\cdots\!38$$ T^{5} + $$80\!\cdots\!63$$ T^{6} + $$23\!\cdots\!38$$ p^{5} T^{7} + 1541935343094270589 p^{10} T^{8} + 32764270404244 p^{15} T^{9} + 1787665859 p^{20} T^{10} + 22326 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	59	$ 1 - 5625510046 T^{2} + 15505742527393176045 T^{4} - $$27\!\cdots\!78$$ T^{6} + $$36\!\cdots\!70$$ T^{8} - $$37\!\cdots\!18$$ T^{10} + $$29\!\cdots\!33$$ T^{12} - $$37\!\cdots\!18$$ p^{10} T^{14} + $$36\!\cdots\!70$$ p^{20} T^{16} - $$27\!\cdots\!78$$ p^{30} T^{18} + 15505742527393176045 p^{40} T^{20} - 5625510046 p^{50} T^{22} + p^{60} T^{24} $
	61	$ ( 1 + 68204 T + 5614052356 T^{2} + 249120175624274 T^{3} + 12083767206480375752 T^{4} + $$39\!\cdots\!68$$ T^{5} + $$13\!\cdots\!51$$ T^{6} + $$39\!\cdots\!68$$ p^{5} T^{7} + 12083767206480375752 p^{10} T^{8} + 249120175624274 p^{15} T^{9} + 5614052356 p^{20} T^{10} + 68204 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	67	$ 1 - 7350772150 T^{2} + 30015657607535269871 T^{4} - $$87\!\cdots\!40$$ T^{6} + $$19\!\cdots\!97$$ T^{8} - $$35\!\cdots\!50$$ T^{10} + $$53\!\cdots\!83$$ T^{12} - $$35\!\cdots\!50$$ p^{10} T^{14} + $$19\!\cdots\!97$$ p^{20} T^{16} - $$87\!\cdots\!40$$ p^{30} T^{18} + 30015657607535269871 p^{40} T^{20} - 7350772150 p^{50} T^{22} + p^{60} T^{24} $
	71	$ 1 - 4774597258 T^{2} + 12520702858599514575 T^{4} - $$31\!\cdots\!60$$ T^{6} + $$79\!\cdots\!49$$ T^{8} - $$15\!\cdots\!26$$ T^{10} + $$26\!\cdots\!19$$ T^{12} - $$15\!\cdots\!26$$ p^{10} T^{14} + $$79\!\cdots\!49$$ p^{20} T^{16} - $$31\!\cdots\!60$$ p^{30} T^{18} + 12520702858599514575 p^{40} T^{20} - 4774597258 p^{50} T^{22} + p^{60} T^{24} $
	73	$ 1 - 9552593482 T^{2} + 48177285345398086899 T^{4} - $$18\!\cdots\!40$$ T^{6} + $$57\!\cdots\!21$$ T^{8} - $$15\!\cdots\!66$$ T^{10} + $$33\!\cdots\!35$$ T^{12} - $$15\!\cdots\!66$$ p^{10} T^{14} + $$57\!\cdots\!21$$ p^{20} T^{16} - $$18\!\cdots\!40$$ p^{30} T^{18} + 48177285345398086899 p^{40} T^{20} - 9552593482 p^{50} T^{22} + p^{60} T^{24} $
	79	$ ( 1 + 66206 T + 4544241147 T^{2} + 483703469494 T^{3} + 5641007475847372879 T^{4} + $$14\!\cdots\!64$$ T^{5} + $$50\!\cdots\!53$$ T^{6} + $$14\!\cdots\!64$$ p^{5} T^{7} + 5641007475847372879 p^{10} T^{8} + 483703469494 p^{15} T^{9} + 4544241147 p^{20} T^{10} + 66206 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	83	$ 1 - 11195217826 T^{2} + 99208519853795290879 T^{4} - $$63\!\cdots\!60$$ T^{6} + $$34\!\cdots\!65$$ T^{8} - $$16\!\cdots\!90$$ T^{10} + $$67\!\cdots\!71$$ T^{12} - $$16\!\cdots\!90$$ p^{10} T^{14} + $$34\!\cdots\!65$$ p^{20} T^{16} - $$63\!\cdots\!60$$ p^{30} T^{18} + 99208519853795290879 p^{40} T^{20} - 11195217826 p^{50} T^{22} + p^{60} T^{24} $
	89	$ 1 - 39744869562 T^{2} + $$74\!\cdots\!83$$ T^{4} - $$88\!\cdots\!96$$ T^{6} + $$74\!\cdots\!81$$ T^{8} - $$49\!\cdots\!62$$ T^{10} + $$28\!\cdots\!51$$ T^{12} - $$49\!\cdots\!62$$ p^{10} T^{14} + $$74\!\cdots\!81$$ p^{20} T^{16} - $$88\!\cdots\!96$$ p^{30} T^{18} + $$74\!\cdots\!83$$ p^{40} T^{20} - 39744869562 p^{50} T^{22} + p^{60} T^{24} $
	97	$ 1 - 46890378730 T^{2} + $$11\!\cdots\!91$$ T^{4} - $$18\!\cdots\!96$$ T^{6} + $$22\!\cdots\!33$$ T^{8} - $$22\!\cdots\!90$$ T^{10} + $$20\!\cdots\!43$$ T^{12} - $$22\!\cdots\!90$$ p^{10} T^{14} + $$22\!\cdots\!33$$ p^{20} T^{16} - $$18\!\cdots\!96$$ p^{30} T^{18} + $$11\!\cdots\!91$$ p^{40} T^{20} - 46890378730 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

−2.96645480635832094116647350863, −2.95269351849685868150387010381, −2.88248654514464903411341706473, −2.69766364114234477399900453727, −2.68910260661456744412751390563, −2.60329493881020499147359156422, −2.27982005608374476364554169900, −2.24905047136271101041411470925, −2.15859646210862122855225398426, −1.92224117907262701330698405021, −1.49896563525021399970706860437, −1.34198845724976464221997413310, −1.30466601500246473987225316950, −1.22914563203969532833643333119, −1.12685071667879053083840302596, −1.10171756134007784484192969754, −0.943246242829028352767280161751, −0.76524932317131826944508084362, −0.67795552097630002957149073389, −0.59866564152673520650368664728, −0.59014141769556175699904993578, −0.50951498449683050153812740499, −0.34808911513049686040642931012, −0.21494319667386406552196360088, −0.15474624214874617988940589715, 0.15474624214874617988940589715, 0.21494319667386406552196360088, 0.34808911513049686040642931012, 0.50951498449683050153812740499, 0.59014141769556175699904993578, 0.59866564152673520650368664728, 0.67795552097630002957149073389, 0.76524932317131826944508084362, 0.943246242829028352767280161751, 1.10171756134007784484192969754, 1.12685071667879053083840302596, 1.22914563203969532833643333119, 1.30466601500246473987225316950, 1.34198845724976464221997413310, 1.49896563525021399970706860437, 1.92224117907262701330698405021, 2.15859646210862122855225398426, 2.24905047136271101041411470925, 2.27982005608374476364554169900, 2.60329493881020499147359156422, 2.68910260661456744412751390563, 2.69766364114234477399900453727, 2.88248654514464903411341706473, 2.95269351849685868150387010381, 2.96645480635832094116647350863

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(0.9538437019\)
\(L(\frac12)\)	\(\approx\)	\(0.9538437019\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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