LMFDB - L-function 12-76e6-1.1-c7e6-0-0

Dirichlet series

L(s) = 1

+ 40·3^-s + 279·5^-s − 1.56e3·7^-s − 3.81e3·9^-s + 7.98e3·11^-s + 1.25e3·13^-s + 1.11e4·15^-s + 2.17e4·17^-s + 4.11e4·19^-s − 6.26e4·21^-s + 1.00e5·23^-s − 8.80e3·25^-s − 2.46e4·27^-s − 5.86e4·29^-s + 4.03e5·31^-s + 3.19e5·33^-s − 4.36e5·35^-s + 8.08e5·37^-s + 5.00e4·39^-s + 5.56e5·41^-s + 1.22e6·43^-s − 1.06e6·45^-s + 1.91e6·47^-s − 2.23e5·49^-s + 8.69e5·51^-s + 5.11e5·53^-s + 2.22e6·55^-s + ⋯

L(s) = 1

+ 0.855·3^-s + 0.998·5^-s − 1.72·7^-s − 1.74·9^-s + 1.80·11^-s + 0.157·13^-s + 0.853·15^-s + 1.07·17^-s + 1.37·19^-s − 1.47·21^-s + 1.72·23^-s − 0.112·25^-s − 0.241·27^-s − 0.446·29^-s + 2.43·31^-s + 1.54·33^-s − 1.72·35^-s + 2.62·37^-s + 0.134·39^-s + 1.26·41^-s + 2.34·43^-s − 1.73·45^-s + 2.69·47^-s − 0.270·49^-s + 0.917·51^-s + 0.472·53^-s + 1.80·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$12$
Conductor:	$2^{12} \cdot 19^{6}$
Sign:	$1$
Analytic conductor:	$1.79069\times 10^{8}$
Root analytic conductor:	$4.87250$
Motivic weight:	$7$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(12,\ 2^{12} \cdot 19^{6} ,\ ( \ : [7/2]^{6} ),\ 1 )$

Particular Values

$L(4)$	$\approx$	$21.24756649$
$L(\frac12)$	$\approx$	$21.24756649$
$L(\frac{9}{2})$		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 $
	19	$ ( 1 - p^{3} T )^{6} $
good	3	$ 1 - 40 T + 5411 T^{2} - 344210 T^{3} + 6991901 p T^{4} - 139790290 p^{2} T^{5} + 1981289758 p^{3} T^{6} - 139790290 p^{9} T^{7} + 6991901 p^{15} T^{8} - 344210 p^{21} T^{9} + 5411 p^{28} T^{10} - 40 p^{35} T^{11} + p^{42} T^{12} $
	5	$ 1 - 279 T + 86643 T^{2} - 19645164 T^{3} + 719997309 p T^{4} - 50853044601 p^{2} T^{5} + 5063034158822 p^{3} T^{6} - 50853044601 p^{9} T^{7} + 719997309 p^{15} T^{8} - 19645164 p^{21} T^{9} + 86643 p^{28} T^{10} - 279 p^{35} T^{11} + p^{42} T^{12} $
	7	$ 1 + 1565 T + 2672300 T^{2} + 2813863905 T^{3} + 3977972497228 T^{4} + 3738876959850925 T^{5} + 4059409837612988806 T^{6} + 3738876959850925 p^{7} T^{7} + 3977972497228 p^{14} T^{8} + 2813863905 p^{21} T^{9} + 2672300 p^{28} T^{10} + 1565 p^{35} T^{11} + p^{42} T^{12} $
	11	$ 1 - 7983 T + 79155245 T^{2} - 344914163226 T^{3} + 1872143133442003 T^{4} - 420087653519920569 p T^{5} + $$27\!\cdots\!02$$ T^{6} - 420087653519920569 p^{8} T^{7} + 1872143133442003 p^{14} T^{8} - 344914163226 p^{21} T^{9} + 79155245 p^{28} T^{10} - 7983 p^{35} T^{11} + p^{42} T^{12} $
	13	$ 1 - 1250 T + 254204373 T^{2} - 398722809000 T^{3} + 31679950959139771 T^{4} - 3835371453362245910 p T^{5} + $$24\!\cdots\!90$$ T^{6} - 3835371453362245910 p^{8} T^{7} + 31679950959139771 p^{14} T^{8} - 398722809000 p^{21} T^{9} + 254204373 p^{28} T^{10} - 1250 p^{35} T^{11} + p^{42} T^{12} $
	17	$ 1 - 21735 T + 1297723454 T^{2} - 7847582348805 T^{3} + 382716495376161820 T^{4} + $$77\!\cdots\!25$$ T^{5} + $$23\!\cdots\!08$$ T^{6} + $$77\!\cdots\!25$$ p^{7} T^{7} + 382716495376161820 p^{14} T^{8} - 7847582348805 p^{21} T^{9} + 1297723454 p^{28} T^{10} - 21735 p^{35} T^{11} + p^{42} T^{12} $
	23	$ 1 - 100920 T + 20144641481 T^{2} - 1566830611331520 T^{3} + $$17\!\cdots\!75$$ T^{4} - $$10\!\cdots\!00$$ T^{5} + $$77\!\cdots\!74$$ T^{6} - $$10\!\cdots\!00$$ p^{7} T^{7} + $$17\!\cdots\!75$$ p^{14} T^{8} - 1566830611331520 p^{21} T^{9} + 20144641481 p^{28} T^{10} - 100920 p^{35} T^{11} + p^{42} T^{12} $
	29	$ 1 + 58656 T + 7787355525 T^{2} + 564070136044080 T^{3} + $$43\!\cdots\!79$$ T^{4} + $$18\!\cdots\!24$$ T^{5} + $$14\!\cdots\!74$$ T^{6} + $$18\!\cdots\!24$$ p^{7} T^{7} + $$43\!\cdots\!79$$ p^{14} T^{8} + 564070136044080 p^{21} T^{9} + 7787355525 p^{28} T^{10} + 58656 p^{35} T^{11} + p^{42} T^{12} $
	31	$ 1 - 403808 T + 183431749962 T^{2} - 51429012710603808 T^{3} + $$13\!\cdots\!31$$ T^{4} - $$27\!\cdots\!44$$ T^{5} + $$48\!\cdots\!16$$ T^{6} - $$27\!\cdots\!44$$ p^{7} T^{7} + $$13\!\cdots\!31$$ p^{14} T^{8} - 51429012710603808 p^{21} T^{9} + 183431749962 p^{28} T^{10} - 403808 p^{35} T^{11} + p^{42} T^{12} $
	37	$ 1 - 808780 T + 20158051586 p T^{2} - 392964738382607340 T^{3} + $$20\!\cdots\!19$$ T^{4} - $$75\!\cdots\!20$$ T^{5} + $$26\!\cdots\!28$$ T^{6} - $$75\!\cdots\!20$$ p^{7} T^{7} + $$20\!\cdots\!19$$ p^{14} T^{8} - 392964738382607340 p^{21} T^{9} + 20158051586 p^{29} T^{10} - 808780 p^{35} T^{11} + p^{42} T^{12} $
	41	$ 1 - 13584 p T + 842468930078 T^{2} - 367901854557576240 T^{3} + $$31\!\cdots\!07$$ T^{4} - $$11\!\cdots\!76$$ T^{5} + $$74\!\cdots\!84$$ T^{6} - $$11\!\cdots\!76$$ p^{7} T^{7} + $$31\!\cdots\!07$$ p^{14} T^{8} - 367901854557576240 p^{21} T^{9} + 842468930078 p^{28} T^{10} - 13584 p^{36} T^{11} + p^{42} T^{12} $
	43	$ 1 - 1220735 T + 1789847433669 T^{2} - 1516343477860082430 T^{3} + $$12\!\cdots\!51$$ T^{4} - $$78\!\cdots\!95$$ T^{5} + $$44\!\cdots\!94$$ T^{6} - $$78\!\cdots\!95$$ p^{7} T^{7} + $$12\!\cdots\!51$$ p^{14} T^{8} - 1516343477860082430 p^{21} T^{9} + 1789847433669 p^{28} T^{10} - 1220735 p^{35} T^{11} + p^{42} T^{12} $
	47	$ 1 - 1915305 T + 2655656379941 T^{2} - 2701214572729595550 T^{3} + $$24\!\cdots\!15$$ T^{4} - $$19\!\cdots\!45$$ T^{5} + $$14\!\cdots\!82$$ T^{6} - $$19\!\cdots\!45$$ p^{7} T^{7} + $$24\!\cdots\!15$$ p^{14} T^{8} - 2701214572729595550 p^{21} T^{9} + 2655656379941 p^{28} T^{10} - 1915305 p^{35} T^{11} + p^{42} T^{12} $
	53	$ 1 - 511650 T + 3307629824861 T^{2} - 1968954407634024000 T^{3} + $$70\!\cdots\!47$$ T^{4} - $$35\!\cdots\!90$$ T^{5} + $$10\!\cdots\!90$$ T^{6} - $$35\!\cdots\!90$$ p^{7} T^{7} + $$70\!\cdots\!47$$ p^{14} T^{8} - 1968954407634024000 p^{21} T^{9} + 3307629824861 p^{28} T^{10} - 511650 p^{35} T^{11} + p^{42} T^{12} $
	59	$ 1 - 1300572 T + 13383936834587 T^{2} - 240978513093007782 p T^{3} + $$78\!\cdots\!75$$ T^{4} - $$66\!\cdots\!82$$ T^{5} + $$25\!\cdots\!98$$ T^{6} - $$66\!\cdots\!82$$ p^{7} T^{7} + $$78\!\cdots\!75$$ p^{14} T^{8} - 240978513093007782 p^{22} T^{9} + 13383936834587 p^{28} T^{10} - 1300572 p^{35} T^{11} + p^{42} T^{12} $
	61	$ 1 - 565335 T + 10120738319115 T^{2} - 5318475573785836408 T^{3} + $$82\!\cdots\!05$$ p T^{4} - $$25\!\cdots\!45$$ T^{5} + $$17\!\cdots\!38$$ T^{6} - $$25\!\cdots\!45$$ p^{7} T^{7} + $$82\!\cdots\!05$$ p^{15} T^{8} - 5318475573785836408 p^{21} T^{9} + 10120738319115 p^{28} T^{10} - 565335 p^{35} T^{11} + p^{42} T^{12} $
	67	$ 1 + 45010 T + 19893322568571 T^{2} + 2865845138838640470 T^{3} + $$23\!\cdots\!31$$ T^{4} + $$31\!\cdots\!40$$ T^{5} + $$17\!\cdots\!66$$ T^{6} + $$31\!\cdots\!40$$ p^{7} T^{7} + $$23\!\cdots\!31$$ p^{14} T^{8} + 2865845138838640470 p^{21} T^{9} + 19893322568571 p^{28} T^{10} + 45010 p^{35} T^{11} + p^{42} T^{12} $
	71	$ 1 + 1424106 T + 40418570081954 T^{2} + 46033418838784453998 T^{3} + $$76\!\cdots\!59$$ T^{4} + $$71\!\cdots\!76$$ T^{5} + $$87\!\cdots\!36$$ T^{6} + $$71\!\cdots\!76$$ p^{7} T^{7} + $$76\!\cdots\!59$$ p^{14} T^{8} + 46033418838784453998 p^{21} T^{9} + 40418570081954 p^{28} T^{10} + 1424106 p^{35} T^{11} + p^{42} T^{12} $
	73	$ 1 + 11153825 T + 112420865725058 T^{2} + $$70\!\cdots\!35$$ T^{3} + $$39\!\cdots\!52$$ T^{4} + $$16\!\cdots\!85$$ T^{5} + $$62\!\cdots\!16$$ T^{6} + $$16\!\cdots\!85$$ p^{7} T^{7} + $$39\!\cdots\!52$$ p^{14} T^{8} + $$70\!\cdots\!35$$ p^{21} T^{9} + 112420865725058 p^{28} T^{10} + 11153825 p^{35} T^{11} + p^{42} T^{12} $
	79	$ 1 + 6392144 T + 118048164146294 T^{2} + $$57\!\cdots\!60$$ T^{3} + $$56\!\cdots\!95$$ T^{4} + $$21\!\cdots\!44$$ T^{5} + $$14\!\cdots\!16$$ T^{6} + $$21\!\cdots\!44$$ p^{7} T^{7} + $$56\!\cdots\!95$$ p^{14} T^{8} + $$57\!\cdots\!60$$ p^{21} T^{9} + 118048164146294 p^{28} T^{10} + 6392144 p^{35} T^{11} + p^{42} T^{12} $
	83	$ 1 + 3164160 T + 59613618243566 T^{2} + $$30\!\cdots\!60$$ T^{3} + $$27\!\cdots\!35$$ T^{4} + $$11\!\cdots\!20$$ T^{5} + $$94\!\cdots\!64$$ T^{6} + $$11\!\cdots\!20$$ p^{7} T^{7} + $$27\!\cdots\!35$$ p^{14} T^{8} + $$30\!\cdots\!60$$ p^{21} T^{9} + 59613618243566 p^{28} T^{10} + 3164160 p^{35} T^{11} + p^{42} T^{12} $
	89	$ 1 + 14502678 T + 153305978961542 T^{2} + $$98\!\cdots\!70$$ T^{3} + $$80\!\cdots\!27$$ T^{4} + $$58\!\cdots\!72$$ T^{5} + $$47\!\cdots\!76$$ T^{6} + $$58\!\cdots\!72$$ p^{7} T^{7} + $$80\!\cdots\!27$$ p^{14} T^{8} + $$98\!\cdots\!70$$ p^{21} T^{9} + 153305978961542 p^{28} T^{10} + 14502678 p^{35} T^{11} + p^{42} T^{12} $
	97	$ 1 + 21377010 T + 439599102471762 T^{2} + $$63\!\cdots\!30$$ T^{3} + $$81\!\cdots\!63$$ T^{4} + $$85\!\cdots\!20$$ T^{5} + $$85\!\cdots\!20$$ T^{6} + $$85\!\cdots\!20$$ p^{7} T^{7} + $$81\!\cdots\!63$$ p^{14} T^{8} + $$63\!\cdots\!30$$ p^{21} T^{9} + 439599102471762 p^{28} T^{10} + 21377010 p^{35} T^{11} + p^{42} T^{12} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−6.67574488800335884529867457285, −6.18657122260240903635275601144, −6.17320406072806216890235118344, −6.04694164979073317308391147589, −5.82553702903268803773392981117, −5.68471044237962129875672287353, −5.58761163664648003385882343442, −5.07397254970131380898691863104, −4.67595830934158854072026646646, −4.56415112347354762337298952788, −4.24542963080829322498855044437, −3.88111449807515538802321559004, −3.79801490038615456046475074895, −3.39095682141689116028038862073, −2.91061033096202832380032045283, −2.83971026732950195504515494061, −2.75397476722405218750834688768, −2.71622296765460085729826411165, −2.44087952178708572018156984996, −1.63821292606797216676389609596, −1.38352089683944589921438345388, −1.13954020283370066529246591284, −0.76093887302894872095068258384, −0.65529836743731433622917632871, −0.44701571400132134004162103608, 0.44701571400132134004162103608, 0.65529836743731433622917632871, 0.76093887302894872095068258384, 1.13954020283370066529246591284, 1.38352089683944589921438345388, 1.63821292606797216676389609596, 2.44087952178708572018156984996, 2.71622296765460085729826411165, 2.75397476722405218750834688768, 2.83971026732950195504515494061, 2.91061033096202832380032045283, 3.39095682141689116028038862073, 3.79801490038615456046475074895, 3.88111449807515538802321559004, 4.24542963080829322498855044437, 4.56415112347354762337298952788, 4.67595830934158854072026646646, 5.07397254970131380898691863104, 5.58761163664648003385882343442, 5.68471044237962129875672287353, 5.82553702903268803773392981117, 6.04694164979073317308391147589, 6.17320406072806216890235118344, 6.18657122260240903635275601144, 6.67574488800335884529867457285

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(21.24756649\)
\(L(\frac12)\)	\(\approx\)	\(21.24756649\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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