LMFDB - L-function 24-21e12-1.1-c6e12-0-0

Dirichlet series

L(s) = 1

+ 52·3^-s + 126·4^-s + 1.03e3·9^-s + 6.55e3·12^-s + 384·13^-s + 1.15e3·16^-s + 1.13e4·19^-s + 5.51e4·25^-s + 4.49e4·27^-s − 9.36e3·31^-s + 1.29e5·36^-s + 2.12e5·37^-s + 1.99e4·39^-s + 2.80e4·43^-s + 6.00e4·48^-s + 1.00e5·49^-s + 4.83e4·52^-s + 5.87e5·57^-s + 9.26e5·61^-s − 6.88e5·64^-s − 2.23e5·67^-s + 6.00e5·73^-s + 2.86e6·75^-s + 1.42e6·76^-s − 2.76e5·79^-s + 2.06e6·81^-s − 4.86e5·93^-s + ⋯

L(s) = 1

+ 1.92·3^-s + 1.96·4^-s + 1.41·9^-s + 3.79·12^-s + 0.174·13^-s + 0.281·16^-s + 1.64·19^-s + 3.52·25^-s + 2.28·27^-s − 0.314·31^-s + 2.78·36^-s + 4.18·37^-s + 0.336·39^-s + 0.353·43^-s + 0.543·48^-s + 6/7·49^-s + 0.344·52^-s + 3.17·57^-s + 4.08·61^-s − 2.62·64^-s − 0.741·67^-s + 1.54·73^-s + 6.79·75^-s + 3.24·76^-s − 0.561·79^-s + 3.88·81^-s − 0.605·93^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$24$
Conductor:	$3^{12} \cdot 7^{12}$
Sign:	$1$
Analytic conductor:	$1.61655\times 10^{8}$
Root analytic conductor:	$2.19798$
Motivic weight:	$6$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(24,\ 3^{12} \cdot 7^{12} ,\ ( \ : [3]^{12} ),\ 1 )$

Particular Values

$L(\frac{7}{2})$	$\approx$	$68.68635815$
$L(\frac12)$	$\approx$	$68.68635815$
$L(4)$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 - 52 T + 62 p^{3} T^{2} - 8716 p^{2} T^{3} + 32435 p^{4} T^{4} - 940480 p^{4} T^{5} + 1056172 p^{7} T^{6} - 940480 p^{10} T^{7} + 32435 p^{16} T^{8} - 8716 p^{20} T^{9} + 62 p^{27} T^{10} - 52 p^{30} T^{11} + p^{36} T^{12} $
	7	$ ( 1 - p^{5} T^{2} )^{6} $
good	2	$ 1 - 63 p T^{2} + 14721 T^{4} - 255221 p^{2} T^{6} + 19824021 p^{2} T^{8} - 7513401 p^{9} T^{10} + 16704875 p^{14} T^{12} - 7513401 p^{21} T^{14} + 19824021 p^{26} T^{16} - 255221 p^{38} T^{18} + 14721 p^{48} T^{20} - 63 p^{61} T^{22} + p^{72} T^{24} $
	5	$ 1 - 55104 T^{2} + 1468600854 T^{4} - 1121223216128 p^{2} T^{6} + 715450128353247 p^{4} T^{8} - 446836833704952768 p^{6} T^{10} + $$28\!\cdots\!56$$ p^{8} T^{12} - 446836833704952768 p^{18} T^{14} + 715450128353247 p^{28} T^{16} - 1121223216128 p^{38} T^{18} + 1468600854 p^{48} T^{20} - 55104 p^{60} T^{22} + p^{72} T^{24} $
	11	$ 1 - 11693640 T^{2} + 60395638769214 T^{4} - $$18\!\cdots\!40$$ T^{6} + $$39\!\cdots\!27$$ T^{8} - $$70\!\cdots\!92$$ T^{10} + $$12\!\cdots\!44$$ T^{12} - $$70\!\cdots\!92$$ p^{12} T^{14} + $$39\!\cdots\!27$$ p^{24} T^{16} - $$18\!\cdots\!40$$ p^{36} T^{18} + 60395638769214 p^{48} T^{20} - 11693640 p^{60} T^{22} + p^{72} T^{24} $
	13	$ ( 1 - 192 T + 19221162 T^{2} + 1668836808 T^{3} + 177653178388515 T^{4} + 39139093191394344 T^{5} + $$10\!\cdots\!56$$ T^{6} + 39139093191394344 p^{6} T^{7} + 177653178388515 p^{12} T^{8} + 1668836808 p^{18} T^{9} + 19221162 p^{24} T^{10} - 192 p^{30} T^{11} + p^{36} T^{12} )^{2} $
	17	$ 1 - 124265232 T^{2} + 8075396374929198 T^{4} - $$35\!\cdots\!40$$ T^{6} + $$12\!\cdots\!87$$ T^{8} - $$33\!\cdots\!00$$ T^{10} + $$84\!\cdots\!68$$ T^{12} - $$33\!\cdots\!00$$ p^{12} T^{14} + $$12\!\cdots\!87$$ p^{24} T^{16} - $$35\!\cdots\!40$$ p^{36} T^{18} + 8075396374929198 p^{48} T^{20} - 124265232 p^{60} T^{22} + p^{72} T^{24} $
	19	$ ( 1 - 5652 T + 179972934 T^{2} - 897430944636 T^{3} + 14455190511039315 T^{4} - 65727793020462160176 T^{5} + $$40\!\cdots\!68$$ p T^{6} - 65727793020462160176 p^{6} T^{7} + 14455190511039315 p^{12} T^{8} - 897430944636 p^{18} T^{9} + 179972934 p^{24} T^{10} - 5652 p^{30} T^{11} + p^{36} T^{12} )^{2} $
	23	$ 1 - 928267704 T^{2} + 456924799440799326 T^{4} - $$15\!\cdots\!16$$ T^{6} + $$39\!\cdots\!99$$ T^{8} - $$80\!\cdots\!64$$ T^{10} + $$13\!\cdots\!96$$ T^{12} - $$80\!\cdots\!64$$ p^{12} T^{14} + $$39\!\cdots\!99$$ p^{24} T^{16} - $$15\!\cdots\!16$$ p^{36} T^{18} + 456924799440799326 p^{48} T^{20} - 928267704 p^{60} T^{22} + p^{72} T^{24} $
	29	$ 1 - 3560516076 T^{2} + 6412459540016584386 T^{4} - $$76\!\cdots\!20$$ T^{6} + $$23\!\cdots\!87$$ p T^{8} - $$49\!\cdots\!92$$ T^{10} + $$30\!\cdots\!60$$ T^{12} - $$49\!\cdots\!92$$ p^{12} T^{14} + $$23\!\cdots\!87$$ p^{25} T^{16} - $$76\!\cdots\!20$$ p^{36} T^{18} + 6412459540016584386 p^{48} T^{20} - 3560516076 p^{60} T^{22} + p^{72} T^{24} $
	31	$ ( 1 + 4680 T + 3236254218 T^{2} + 31562116597576 T^{3} + 5286868070613701343 T^{4} + $$60\!\cdots\!04$$ T^{5} + $$56\!\cdots\!84$$ T^{6} + $$60\!\cdots\!04$$ p^{6} T^{7} + 5286868070613701343 p^{12} T^{8} + 31562116597576 p^{18} T^{9} + 3236254218 p^{24} T^{10} + 4680 p^{30} T^{11} + p^{36} T^{12} )^{2} $
	37	$ ( 1 - 106008 T + 17251806222 T^{2} - 34202925512408 p T^{3} + $$11\!\cdots\!03$$ T^{4} - $$62\!\cdots\!84$$ T^{5} + $$39\!\cdots\!16$$ T^{6} - $$62\!\cdots\!84$$ p^{6} T^{7} + $$11\!\cdots\!03$$ p^{12} T^{8} - 34202925512408 p^{19} T^{9} + 17251806222 p^{24} T^{10} - 106008 p^{30} T^{11} + p^{36} T^{12} )^{2} $
	41	$ 1 - 39047495184 T^{2} + $$73\!\cdots\!82$$ T^{4} - $$87\!\cdots\!04$$ T^{6} + $$75\!\cdots\!87$$ T^{8} - $$50\!\cdots\!16$$ T^{10} + $$26\!\cdots\!64$$ T^{12} - $$50\!\cdots\!16$$ p^{12} T^{14} + $$75\!\cdots\!87$$ p^{24} T^{16} - $$87\!\cdots\!04$$ p^{36} T^{18} + $$73\!\cdots\!82$$ p^{48} T^{20} - 39047495184 p^{60} T^{22} + p^{72} T^{24} $
	43	$ ( 1 - 14040 T + 9749002014 T^{2} + 303502458318728 T^{3} + 83456374862616461487 T^{4} + $$11\!\cdots\!20$$ T^{5} + $$68\!\cdots\!44$$ T^{6} + $$11\!\cdots\!20$$ p^{6} T^{7} + 83456374862616461487 p^{12} T^{8} + 303502458318728 p^{18} T^{9} + 9749002014 p^{24} T^{10} - 14040 p^{30} T^{11} + p^{36} T^{12} )^{2} $
	47	$ 1 - 49641718140 T^{2} + $$11\!\cdots\!42$$ T^{4} - $$14\!\cdots\!84$$ T^{6} + $$93\!\cdots\!71$$ T^{8} + $$28\!\cdots\!04$$ T^{10} - $$10\!\cdots\!44$$ T^{12} + $$28\!\cdots\!04$$ p^{12} T^{14} + $$93\!\cdots\!71$$ p^{24} T^{16} - $$14\!\cdots\!84$$ p^{36} T^{18} + $$11\!\cdots\!42$$ p^{48} T^{20} - 49641718140 p^{60} T^{22} + p^{72} T^{24} $
	53	$ 1 - 161105116284 T^{2} + $$12\!\cdots\!30$$ T^{4} - $$58\!\cdots\!04$$ T^{6} + $$19\!\cdots\!35$$ T^{8} - $$54\!\cdots\!32$$ T^{10} + $$12\!\cdots\!52$$ T^{12} - $$54\!\cdots\!32$$ p^{12} T^{14} + $$19\!\cdots\!35$$ p^{24} T^{16} - $$58\!\cdots\!04$$ p^{36} T^{18} + $$12\!\cdots\!30$$ p^{48} T^{20} - 161105116284 p^{60} T^{22} + p^{72} T^{24} $
	59	$ 1 - 337822020396 T^{2} + $$56\!\cdots\!70$$ T^{4} - $$62\!\cdots\!48$$ T^{6} + $$50\!\cdots\!23$$ T^{8} - $$30\!\cdots\!20$$ T^{10} + $$14\!\cdots\!04$$ T^{12} - $$30\!\cdots\!20$$ p^{12} T^{14} + $$50\!\cdots\!23$$ p^{24} T^{16} - $$62\!\cdots\!48$$ p^{36} T^{18} + $$56\!\cdots\!70$$ p^{48} T^{20} - 337822020396 p^{60} T^{22} + p^{72} T^{24} $
	61	$ ( 1 - 463368 T + 274969916586 T^{2} - 100295001713623824 T^{3} + $$34\!\cdots\!15$$ T^{4} - $$93\!\cdots\!16$$ T^{5} + $$23\!\cdots\!24$$ T^{6} - $$93\!\cdots\!16$$ p^{6} T^{7} + $$34\!\cdots\!15$$ p^{12} T^{8} - 100295001713623824 p^{18} T^{9} + 274969916586 p^{24} T^{10} - 463368 p^{30} T^{11} + p^{36} T^{12} )^{2} $
	67	$ ( 1 + 111552 T + 143020276686 T^{2} - 3395120375363616 T^{3} + $$81\!\cdots\!19$$ T^{4} - $$14\!\cdots\!68$$ T^{5} + $$62\!\cdots\!12$$ T^{6} - $$14\!\cdots\!68$$ p^{6} T^{7} + $$81\!\cdots\!19$$ p^{12} T^{8} - 3395120375363616 p^{18} T^{9} + 143020276686 p^{24} T^{10} + 111552 p^{30} T^{11} + p^{36} T^{12} )^{2} $
	71	$ 1 - 401361224808 T^{2} + $$12\!\cdots\!18$$ T^{4} - $$27\!\cdots\!04$$ T^{6} + $$51\!\cdots\!67$$ T^{8} - $$79\!\cdots\!76$$ T^{10} + $$11\!\cdots\!08$$ T^{12} - $$79\!\cdots\!76$$ p^{12} T^{14} + $$51\!\cdots\!67$$ p^{24} T^{16} - $$27\!\cdots\!04$$ p^{36} T^{18} + $$12\!\cdots\!18$$ p^{48} T^{20} - 401361224808 p^{60} T^{22} + p^{72} T^{24} $
	73	$ ( 1 - 300492 T + 872431102194 T^{2} - 211773247621274012 T^{3} + $$32\!\cdots\!79$$ T^{4} - $$61\!\cdots\!24$$ T^{5} + $$64\!\cdots\!00$$ T^{6} - $$61\!\cdots\!24$$ p^{6} T^{7} + $$32\!\cdots\!79$$ p^{12} T^{8} - 211773247621274012 p^{18} T^{9} + 872431102194 p^{24} T^{10} - 300492 p^{30} T^{11} + p^{36} T^{12} )^{2} $
	79	$ ( 1 + 138432 T + 745227854070 T^{2} + 239274317465345984 T^{3} + $$28\!\cdots\!87$$ T^{4} + $$12\!\cdots\!72$$ T^{5} + $$74\!\cdots\!12$$ T^{6} + $$12\!\cdots\!72$$ p^{6} T^{7} + $$28\!\cdots\!87$$ p^{12} T^{8} + 239274317465345984 p^{18} T^{9} + 745227854070 p^{24} T^{10} + 138432 p^{30} T^{11} + p^{36} T^{12} )^{2} $
	83	$ 1 - 2422223724828 T^{2} + $$25\!\cdots\!02$$ T^{4} - $$15\!\cdots\!28$$ T^{6} + $$59\!\cdots\!39$$ T^{8} - $$15\!\cdots\!60$$ T^{10} + $$38\!\cdots\!20$$ T^{12} - $$15\!\cdots\!60$$ p^{12} T^{14} + $$59\!\cdots\!39$$ p^{24} T^{16} - $$15\!\cdots\!28$$ p^{36} T^{18} + $$25\!\cdots\!02$$ p^{48} T^{20} - 2422223724828 p^{60} T^{22} + p^{72} T^{24} $
	89	$ 1 - 2000094606384 T^{2} + $$28\!\cdots\!10$$ T^{4} - $$28\!\cdots\!32$$ T^{6} + $$22\!\cdots\!11$$ T^{8} - $$14\!\cdots\!44$$ T^{10} + $$81\!\cdots\!52$$ T^{12} - $$14\!\cdots\!44$$ p^{12} T^{14} + $$22\!\cdots\!11$$ p^{24} T^{16} - $$28\!\cdots\!32$$ p^{36} T^{18} + $$28\!\cdots\!10$$ p^{48} T^{20} - 2000094606384 p^{60} T^{22} + p^{72} T^{24} $
	97	$ ( 1 - 810708 T + 2967260359170 T^{2} - 1871491030983620996 T^{3} + $$47\!\cdots\!11$$ p T^{4} - $$24\!\cdots\!60$$ T^{5} + $$46\!\cdots\!16$$ T^{6} - $$24\!\cdots\!60$$ p^{6} T^{7} + $$47\!\cdots\!11$$ p^{13} T^{8} - 1871491030983620996 p^{18} T^{9} + 2967260359170 p^{24} T^{10} - 810708 p^{30} T^{11} + p^{36} T^{12} )^{2} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−5.90122800610158876267459602792, −5.41510531109308055729597557572, −5.35118093214802912043057852106, −5.17751341206966365382958125441, −4.96771715157316624203870195215, −4.84489684295728465501697310484, −4.60098212351708496930235408665, −4.54495670151747176774254494294, −4.17058359341817361858497612376, −3.86643051856995147371921753482, −3.84382803302262106565075646599, −3.61037685467188472779699927690, −3.15508755600966401002801477697, −3.02622517914671722806383721871, −2.99262467366116793143473840113, −2.62505627958476827898364828270, −2.51651143566173053935851017952, −2.44922258492772732693019279014, −2.27323561465292267517968495733, −1.97951011247600285315682796959, −1.33912215542198745202664543877, −1.13113987237378904863084955544, −1.11835804180406885395044860654, −0.59708101761326252455495282298, −0.57204046252275183881874679296, 0.57204046252275183881874679296, 0.59708101761326252455495282298, 1.11835804180406885395044860654, 1.13113987237378904863084955544, 1.33912215542198745202664543877, 1.97951011247600285315682796959, 2.27323561465292267517968495733, 2.44922258492772732693019279014, 2.51651143566173053935851017952, 2.62505627958476827898364828270, 2.99262467366116793143473840113, 3.02622517914671722806383721871, 3.15508755600966401002801477697, 3.61037685467188472779699927690, 3.84382803302262106565075646599, 3.86643051856995147371921753482, 4.17058359341817361858497612376, 4.54495670151747176774254494294, 4.60098212351708496930235408665, 4.84489684295728465501697310484, 4.96771715157316624203870195215, 5.17751341206966365382958125441, 5.35118093214802912043057852106, 5.41510531109308055729597557572, 5.90122800610158876267459602792

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 $\F_{q}$

\(L(\frac{7}{2})\)	\(\approx\)	\(68.68635815\)
\(L(\frac12)\)	\(\approx\)	\(68.68635815\)
\(L(4)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Origins of factors

Downloads

Learn more