LMFDB - L-function 12-74e6-1.1-c7e6-0-0

Dirichlet series

L(s) = 1

− 48·2^-s + 28·3^-s + 1.34e3·4^-s − 14·5^-s − 1.34e3·6^-s − 980·7^-s − 2.86e4·8^-s − 2.04e3·9^-s + 672·10^-s + 2.95e3·11^-s + 3.76e4·12^-s + 2.39e3·13^-s + 4.70e4·14^-s − 392·15^-s + 5.16e5·16^-s − 4.51e4·17^-s + 9.80e4·18^-s + 1.17e4·19^-s − 1.88e4·20^-s − 2.74e4·21^-s − 1.41e5·22^-s + 2.10e4·23^-s − 8.02e5·24^-s − 1.36e5·25^-s − 1.14e5·26^-s + 6.15e4·27^-s − 1.31e6·28^-s + ⋯

L(s) = 1

− 4.24·2^-s + 0.598·3^-s + 21/2·4^-s − 0.0500·5^-s − 2.54·6^-s − 1.07·7^-s − 19.7·8^-s − 0.933·9^-s + 0.212·10^-s + 0.669·11^-s + 6.28·12^-s + 0.302·13^-s + 4.58·14^-s − 0.0299·15^-s + 63/2·16^-s − 2.22·17^-s + 3.96·18^-s + 0.393·19^-s − 0.525·20^-s − 0.646·21^-s − 2.84·22^-s + 0.360·23^-s − 11.8·24^-s − 1.75·25^-s − 1.28·26^-s + 0.601·27^-s − 11.3·28^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$12$
Conductor:	$2^{6} \cdot 37^{6}$
Sign:	$1$
Analytic conductor:	$1.52591\times 10^{8}$
Root analytic conductor:	$4.80796$
Motivic weight:	$7$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(12,\ 2^{6} \cdot 37^{6} ,\ ( \ : [7/2]^{6} ),\ 1 )$

Particular Values

$L(4)$	$\approx$	$0.6366603934$
$L(\frac12)$	$\approx$	$0.6366603934$
$L(\frac{9}{2})$		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 + p^{3} T )^{6} $
	37	$ ( 1 + p^{3} T )^{6} $
good	3	$ 1 - 28 T + 314 p^{2} T^{2} - 197812 T^{3} + 4208045 p T^{4} - 2613508 p^{5} T^{5} + 953704841 p^{3} T^{6} - 2613508 p^{12} T^{7} + 4208045 p^{15} T^{8} - 197812 p^{21} T^{9} + 314 p^{30} T^{10} - 28 p^{35} T^{11} + p^{42} T^{12} $
	5	$ 1 + 14 T + 137101 T^{2} + 38861676 T^{3} + 2249522144 p T^{4} + 960706393734 p T^{5} + 46373133827628 p^{2} T^{6} + 960706393734 p^{8} T^{7} + 2249522144 p^{15} T^{8} + 38861676 p^{21} T^{9} + 137101 p^{28} T^{10} + 14 p^{35} T^{11} + p^{42} T^{12} $
	7	$ 1 + 20 p^{2} T + 1720401 T^{2} + 19852916 p^{2} T^{3} + 1047083812575 T^{4} + 13462029050304 p^{2} T^{5} + 688712107253953166 T^{6} + 13462029050304 p^{9} T^{7} + 1047083812575 p^{14} T^{8} + 19852916 p^{23} T^{9} + 1720401 p^{28} T^{10} + 20 p^{37} T^{11} + p^{42} T^{12} $
	11	$ 1 - 2956 T + 3630002 p T^{2} - 1177454780 T^{3} + 98650641688625 p T^{4} - 93334256315273948 p T^{5} + $$30\!\cdots\!11$$ T^{6} - 93334256315273948 p^{8} T^{7} + 98650641688625 p^{15} T^{8} - 1177454780 p^{21} T^{9} + 3630002 p^{29} T^{10} - 2956 p^{35} T^{11} + p^{42} T^{12} $
	13	$ 1 - 2394 T + 156273485 T^{2} - 597574230116 T^{3} + 13244142724924304 T^{4} - 83993463153336586690 T^{5} + $$89\!\cdots\!24$$ T^{6} - 83993463153336586690 p^{7} T^{7} + 13244142724924304 p^{14} T^{8} - 597574230116 p^{21} T^{9} + 156273485 p^{28} T^{10} - 2394 p^{35} T^{11} + p^{42} T^{12} $
	17	$ 1 + 45108 T + 1693099934 T^{2} + 1797683304996 p T^{3} + 339716846738737439 T^{4} - $$51\!\cdots\!12$$ T^{5} - $$16\!\cdots\!68$$ T^{6} - $$51\!\cdots\!12$$ p^{7} T^{7} + 339716846738737439 p^{14} T^{8} + 1797683304996 p^{22} T^{9} + 1693099934 p^{28} T^{10} + 45108 p^{35} T^{11} + p^{42} T^{12} $
	19	$ 1 - 11764 T + 2591876282 T^{2} - 1612566519364 p T^{3} + 4313445313829093031 T^{4} - $$42\!\cdots\!52$$ T^{5} + $$45\!\cdots\!52$$ T^{6} - $$42\!\cdots\!52$$ p^{7} T^{7} + 4313445313829093031 p^{14} T^{8} - 1612566519364 p^{22} T^{9} + 2591876282 p^{28} T^{10} - 11764 p^{35} T^{11} + p^{42} T^{12} $
	23	$ 1 - 21052 T + 8712586979 T^{2} + 197856719062424 T^{3} + 36630835070672085560 T^{4} + $$14\!\cdots\!68$$ T^{5} + $$15\!\cdots\!28$$ T^{6} + $$14\!\cdots\!68$$ p^{7} T^{7} + 36630835070672085560 p^{14} T^{8} + 197856719062424 p^{21} T^{9} + 8712586979 p^{28} T^{10} - 21052 p^{35} T^{11} + p^{42} T^{12} $
	29	$ 1 - 288454 T + 73405416045 T^{2} - 12973049741078676 T^{3} + $$21\!\cdots\!16$$ T^{4} - $$29\!\cdots\!02$$ T^{5} + $$42\!\cdots\!56$$ T^{6} - $$29\!\cdots\!02$$ p^{7} T^{7} + $$21\!\cdots\!16$$ p^{14} T^{8} - 12973049741078676 p^{21} T^{9} + 73405416045 p^{28} T^{10} - 288454 p^{35} T^{11} + p^{42} T^{12} $
	31	$ 1 - 578868 T + 245248469923 T^{2} - 70229973225888120 T^{3} + $$17\!\cdots\!84$$ T^{4} - $$33\!\cdots\!16$$ T^{5} + $$61\!\cdots\!84$$ T^{6} - $$33\!\cdots\!16$$ p^{7} T^{7} + $$17\!\cdots\!84$$ p^{14} T^{8} - 70229973225888120 p^{21} T^{9} + 245248469923 p^{28} T^{10} - 578868 p^{35} T^{11} + p^{42} T^{12} $
	41	$ 1 - 1176840 T + 1353320825098 T^{2} - 1052175227166895578 T^{3} + $$71\!\cdots\!83$$ T^{4} - $$39\!\cdots\!74$$ T^{5} + $$19\!\cdots\!65$$ T^{6} - $$39\!\cdots\!74$$ p^{7} T^{7} + $$71\!\cdots\!83$$ p^{14} T^{8} - 1052175227166895578 p^{21} T^{9} + 1353320825098 p^{28} T^{10} - 1176840 p^{35} T^{11} + p^{42} T^{12} $
	43	$ 1 - 2669236 T + 4151202419498 T^{2} - 4596969313952877436 T^{3} + $$39\!\cdots\!59$$ T^{4} - $$27\!\cdots\!16$$ T^{5} + $$15\!\cdots\!24$$ T^{6} - $$27\!\cdots\!16$$ p^{7} T^{7} + $$39\!\cdots\!59$$ p^{14} T^{8} - 4596969313952877436 p^{21} T^{9} + 4151202419498 p^{28} T^{10} - 2669236 p^{35} T^{11} + p^{42} T^{12} $
	47	$ 1 + 131044 T + 909771102465 T^{2} - 43660093396560108 T^{3} + $$47\!\cdots\!79$$ T^{4} - $$20\!\cdots\!92$$ T^{5} + $$24\!\cdots\!78$$ T^{6} - $$20\!\cdots\!92$$ p^{7} T^{7} + $$47\!\cdots\!79$$ p^{14} T^{8} - 43660093396560108 p^{21} T^{9} + 909771102465 p^{28} T^{10} + 131044 p^{35} T^{11} + p^{42} T^{12} $
	53	$ 1 - 983190 T + 1677233327675 T^{2} - 1072795661170862726 T^{3} + $$25\!\cdots\!59$$ T^{4} - $$19\!\cdots\!36$$ T^{5} + $$32\!\cdots\!54$$ T^{6} - $$19\!\cdots\!36$$ p^{7} T^{7} + $$25\!\cdots\!59$$ p^{14} T^{8} - 1072795661170862726 p^{21} T^{9} + 1677233327675 p^{28} T^{10} - 983190 p^{35} T^{11} + p^{42} T^{12} $
	59	$ 1 + 1215568 T + 4910616279926 T^{2} + 5700828021613663344 T^{3} + $$21\!\cdots\!99$$ T^{4} + $$20\!\cdots\!08$$ T^{5} + $$57\!\cdots\!76$$ T^{6} + $$20\!\cdots\!08$$ p^{7} T^{7} + $$21\!\cdots\!99$$ p^{14} T^{8} + 5700828021613663344 p^{21} T^{9} + 4910616279926 p^{28} T^{10} + 1215568 p^{35} T^{11} + p^{42} T^{12} $
	61	$ 1 - 3136358 T + 14029027452217 T^{2} - 32087580051924737196 T^{3} + $$80\!\cdots\!56$$ T^{4} - $$15\!\cdots\!98$$ T^{5} + $$29\!\cdots\!56$$ T^{6} - $$15\!\cdots\!98$$ p^{7} T^{7} + $$80\!\cdots\!56$$ p^{14} T^{8} - 32087580051924737196 p^{21} T^{9} + 14029027452217 p^{28} T^{10} - 3136358 p^{35} T^{11} + p^{42} T^{12} $
	67	$ 1 - 2179276 T + 32154658886415 T^{2} - 59551484011079515600 T^{3} + $$45\!\cdots\!76$$ T^{4} - $$68\!\cdots\!12$$ T^{5} + $$35\!\cdots\!52$$ T^{6} - $$68\!\cdots\!12$$ p^{7} T^{7} + $$45\!\cdots\!76$$ p^{14} T^{8} - 59551484011079515600 p^{21} T^{9} + 32154658886415 p^{28} T^{10} - 2179276 p^{35} T^{11} + p^{42} T^{12} $
	71	$ 1 - 325164 T + 31175640012965 T^{2} - 45094243169301280836 T^{3} + $$43\!\cdots\!47$$ T^{4} - $$10\!\cdots\!96$$ T^{5} + $$43\!\cdots\!02$$ T^{6} - $$10\!\cdots\!96$$ p^{7} T^{7} + $$43\!\cdots\!47$$ p^{14} T^{8} - 45094243169301280836 p^{21} T^{9} + 31175640012965 p^{28} T^{10} - 325164 p^{35} T^{11} + p^{42} T^{12} $
	73	$ 1 - 5011444 T + 34194444086858 T^{2} - $$10\!\cdots\!22$$ T^{3} + $$53\!\cdots\!79$$ T^{4} - $$14\!\cdots\!54$$ T^{5} + $$63\!\cdots\!77$$ T^{6} - $$14\!\cdots\!54$$ p^{7} T^{7} + $$53\!\cdots\!79$$ p^{14} T^{8} - $$10\!\cdots\!22$$ p^{21} T^{9} + 34194444086858 p^{28} T^{10} - 5011444 p^{35} T^{11} + p^{42} T^{12} $
	79	$ 1 - 3173032 T + 98982451603591 T^{2} - $$27\!\cdots\!64$$ T^{3} + $$43\!\cdots\!56$$ T^{4} - $$10\!\cdots\!48$$ T^{5} + $$10\!\cdots\!68$$ T^{6} - $$10\!\cdots\!48$$ p^{7} T^{7} + $$43\!\cdots\!56$$ p^{14} T^{8} - $$27\!\cdots\!64$$ p^{21} T^{9} + 98982451603591 p^{28} T^{10} - 3173032 p^{35} T^{11} + p^{42} T^{12} $
	83	$ 1 + 22567048 T + 340865624480273 T^{2} + $$35\!\cdots\!68$$ T^{3} + $$30\!\cdots\!87$$ T^{4} + $$20\!\cdots\!52$$ T^{5} + $$11\!\cdots\!14$$ T^{6} + $$20\!\cdots\!52$$ p^{7} T^{7} + $$30\!\cdots\!87$$ p^{14} T^{8} + $$35\!\cdots\!68$$ p^{21} T^{9} + 340865624480273 p^{28} T^{10} + 22567048 p^{35} T^{11} + p^{42} T^{12} $
	89	$ 1 - 26836996 T + 436727416452614 T^{2} - $$51\!\cdots\!40$$ T^{3} + $$49\!\cdots\!03$$ T^{4} - $$41\!\cdots\!04$$ T^{5} + $$29\!\cdots\!88$$ T^{6} - $$41\!\cdots\!04$$ p^{7} T^{7} + $$49\!\cdots\!03$$ p^{14} T^{8} - $$51\!\cdots\!40$$ p^{21} T^{9} + 436727416452614 p^{28} T^{10} - 26836996 p^{35} T^{11} + p^{42} T^{12} $
	97	$ 1 - 295792 T + 260268831210958 T^{2} - $$10\!\cdots\!80$$ T^{3} + $$29\!\cdots\!43$$ T^{4} - $$23\!\cdots\!48$$ T^{5} + $$24\!\cdots\!44$$ T^{6} - $$23\!\cdots\!48$$ p^{7} T^{7} + $$29\!\cdots\!43$$ p^{14} T^{8} - $$10\!\cdots\!80$$ p^{21} T^{9} + 260268831210958 p^{28} T^{10} - 295792 p^{35} T^{11} + p^{42} T^{12} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−6.70396101778551677674977560775, −6.45934762301571968344495601521, −6.43107077810357683107815789033, −6.33944329516245058532008894079, −6.21661975984554573517147321830, −6.04422340256838188417480087770, −5.60993430612216576848305701979, −5.21380588695754204657225482131, −4.83498105484018947045088556477, −4.58670570594933045740155239058, −4.10187065682825513668347355537, −3.80046991308858061056648105267, −3.77905155872665571343701753993, −3.14087168157851652116598532855, −2.72001643768194291546875366630, −2.64778034547260045751776077716, −2.54499302609837275849153217123, −2.35997394871046904638773156340, −2.18921883059406607295567202367, −1.36003616208743880172402456756, −1.28407571223361304310102957593, −0.933868402556889095920275503668, −0.69321964202088485545760055001, −0.56121230674389349300985486201, −0.24193973982999600292923148944, 0.24193973982999600292923148944, 0.56121230674389349300985486201, 0.69321964202088485545760055001, 0.933868402556889095920275503668, 1.28407571223361304310102957593, 1.36003616208743880172402456756, 2.18921883059406607295567202367, 2.35997394871046904638773156340, 2.54499302609837275849153217123, 2.64778034547260045751776077716, 2.72001643768194291546875366630, 3.14087168157851652116598532855, 3.77905155872665571343701753993, 3.80046991308858061056648105267, 4.10187065682825513668347355537, 4.58670570594933045740155239058, 4.83498105484018947045088556477, 5.21380588695754204657225482131, 5.60993430612216576848305701979, 6.04422340256838188417480087770, 6.21661975984554573517147321830, 6.33944329516245058532008894079, 6.43107077810357683107815789033, 6.45934762301571968344495601521, 6.70396101778551677674977560775

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(0.6366603934\)
\(L(\frac12)\)	\(\approx\)	\(0.6366603934\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Origins of factors

Downloads

Learn more