LMFDB - L-function 4-3e4-1.1-c37e2-0-0

Dirichlet series

L(s) = 1

+ 1.94e5·2^-s − 9.96e10·4^-s − 5.52e12·5^-s − 3.44e15·7^-s − 1.93e16·8^-s − 1.07e18·10^-s + 2.67e19·11^-s + 5.30e20·13^-s − 6.70e20·14^-s − 3.76e21·16^-s + 8.94e22·17^-s + 3.73e23·19^-s + 5.51e23·20^-s + 5.19e24·22^-s + 2.62e25·23^-s − 2.20e23·25^-s + 1.03e26·26^-s + 3.43e26·28^-s + 1.27e27·29^-s + 2.61e26·31^-s − 4.40e27·32^-s + 1.73e28·34^-s + 1.90e28·35^-s − 6.80e28·37^-s + 7.25e28·38^-s + 1.07e29·40^-s + 1.26e30·41^-s + ⋯

L(s) = 1

+ 0.524·2^-s − 0.725·4^-s − 0.648·5^-s − 0.800·7^-s − 0.380·8^-s − 0.339·10^-s + 1.44·11^-s + 1.30·13^-s − 0.419·14^-s − 0.199·16^-s + 1.54·17^-s + 0.822·19^-s + 0.470·20^-s + 0.760·22^-s + 1.68·23^-s − 0.00302·25^-s + 0.686·26^-s + 0.580·28^-s + 1.12·29^-s + 0.0671·31^-s − 0.628·32^-s + 0.808·34^-s + 0.518·35^-s − 0.662·37^-s + 0.431·38^-s + 0.246·40^-s + 1.83·41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+37/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$81$ = $3^{4}$
Sign:	$1$
Analytic conductor:	$6090.65$
Root analytic conductor:	$8.83417$
Motivic weight:	$37$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 81,\ (\ :37/2, 37/2),\ 1)$

Particular Values

$L(19)$	$\approx$	$3.654604134$
$L(\frac12)$	$\approx$	$3.654604134$
$L(\frac{39}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$\Gal(F_p)$	$F_p(T)$
bad	3		$ 1 $
good	2	$D_{4}$	$ 1 - 6075 p^{5} T + 16781555 p^{13} T^{2} - 6075 p^{42} T^{3} + p^{74} T^{4} $
	5	$D_{4}$	$ 1 + 221183375436 p^{2} T + $$39\!\cdots\!58$$ p^{7} T^{2} + 221183375436 p^{39} T^{3} + p^{74} T^{4} $
	7	$D_{4}$	$ 1 + 70376407214000 p^{2} T + $$15\!\cdots\!50$$ p^{5} T^{2} + 70376407214000 p^{39} T^{3} + p^{74} T^{4} $
	11	$D_{4}$	$ 1 - 2430366941349867096 p T + $$53\!\cdots\!46$$ p^{3} T^{2} - 2430366941349867096 p^{38} T^{3} + p^{74} T^{4} $
	13	$D_{4}$	$ 1 - 40813980127225629100 p T + $$14\!\cdots\!70$$ p^{3} T^{2} - 40813980127225629100 p^{38} T^{3} + p^{74} T^{4} $
	17	$D_{4}$	$ 1 - $$52\!\cdots\!00$$ p T + $$17\!\cdots\!10$$ p^{3} T^{2} - $$52\!\cdots\!00$$ p^{38} T^{3} + p^{74} T^{4} $
	19	$D_{4}$	$ 1 - $$37\!\cdots\!20$$ T + $$10\!\cdots\!62$$ p T^{2} - $$37\!\cdots\!20$$ p^{37} T^{3} + p^{74} T^{4} $
	23	$D_{4}$	$ 1 - $$26\!\cdots\!00$$ T + $$21\!\cdots\!70$$ p T^{2} - $$26\!\cdots\!00$$ p^{37} T^{3} + p^{74} T^{4} $
	29	$D_{4}$	$ 1 - $$43\!\cdots\!80$$ p T + $$26\!\cdots\!98$$ p^{2} T^{2} - $$43\!\cdots\!80$$ p^{38} T^{3} + p^{74} T^{4} $
	31	$D_{4}$	$ 1 - $$84\!\cdots\!04$$ p T + $$25\!\cdots\!06$$ p^{2} T^{2} - $$84\!\cdots\!04$$ p^{38} T^{3} + p^{74} T^{4} $
	37	$D_{4}$	$ 1 + $$49\!\cdots\!00$$ p^{2} T + $$13\!\cdots\!90$$ T^{2} + $$49\!\cdots\!00$$ p^{39} T^{3} + p^{74} T^{4} $
	41	$D_{4}$	$ 1 - $$12\!\cdots\!36$$ T + $$12\!\cdots\!86$$ T^{2} - $$12\!\cdots\!36$$ p^{37} T^{3} + p^{74} T^{4} $
	43	$D_{4}$	$ 1 + $$25\!\cdots\!00$$ T + $$44\!\cdots\!50$$ T^{2} + $$25\!\cdots\!00$$ p^{37} T^{3} + p^{74} T^{4} $
	47	$D_{4}$	$ 1 + $$42\!\cdots\!00$$ T + $$14\!\cdots\!70$$ T^{2} + $$42\!\cdots\!00$$ p^{37} T^{3} + p^{74} T^{4} $
	53	$D_{4}$	$ 1 + $$15\!\cdots\!00$$ T + $$16\!\cdots\!70$$ T^{2} + $$15\!\cdots\!00$$ p^{37} T^{3} + p^{74} T^{4} $
	59	$D_{4}$	$ 1 - $$23\!\cdots\!40$$ T + $$64\!\cdots\!38$$ T^{2} - $$23\!\cdots\!40$$ p^{37} T^{3} + p^{74} T^{4} $
	61	$D_{4}$	$ 1 - $$10\!\cdots\!44$$ T + $$10\!\cdots\!26$$ T^{2} - $$10\!\cdots\!44$$ p^{37} T^{3} + p^{74} T^{4} $
	67	$D_{4}$	$ 1 + $$10\!\cdots\!00$$ T + $$94\!\cdots\!30$$ T^{2} + $$10\!\cdots\!00$$ p^{37} T^{3} + p^{74} T^{4} $
	71	$D_{4}$	$ 1 - $$74\!\cdots\!16$$ T + $$58\!\cdots\!46$$ T^{2} - $$74\!\cdots\!16$$ p^{37} T^{3} + p^{74} T^{4} $
	73	$D_{4}$	$ 1 - $$19\!\cdots\!00$$ T + $$18\!\cdots\!10$$ T^{2} - $$19\!\cdots\!00$$ p^{37} T^{3} + p^{74} T^{4} $
	79	$D_{4}$	$ 1 - $$27\!\cdots\!80$$ T + $$64\!\cdots\!42$$ p T^{2} - $$27\!\cdots\!80$$ p^{37} T^{3} + p^{74} T^{4} $
	83	$D_{4}$	$ 1 - $$47\!\cdots\!00$$ T + $$24\!\cdots\!30$$ T^{2} - $$47\!\cdots\!00$$ p^{37} T^{3} + p^{74} T^{4} $
	89	$D_{4}$	$ 1 - $$13\!\cdots\!60$$ T + $$23\!\cdots\!58$$ T^{2} - $$13\!\cdots\!60$$ p^{37} T^{3} + p^{74} T^{4} $
	97	$D_{4}$	$ 1 - $$60\!\cdots\!00$$ T + $$57\!\cdots\!70$$ T^{2} - $$60\!\cdots\!00$$ p^{37} T^{3} + p^{74} T^{4} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−13.56982560306490774743633173202, −12.97924981677093867485116166361, −12.32977298346883012399583422873, −11.69793327349586923753835272384, −11.08638761826484467966094767741, −10.18487672862619229966687375147, −9.240843808415960017862893798652, −9.141835712938481320721142610050, −8.181767095037083665039383790276, −7.47818031133602201288462997171, −6.50631012254476021640822163523, −6.23557045964603883301190964202, −5.10942836408317068989441894151, −4.72121089856995631565104750498, −3.63356016971806899270931581536, −3.58817879819539040188777686579, −2.98838845480565957989625613917, −1.55118135216588969185328888964, −1.01378084956321360107102645348, −0.52000855062034144903580596604, 0.52000855062034144903580596604, 1.01378084956321360107102645348, 1.55118135216588969185328888964, 2.98838845480565957989625613917, 3.58817879819539040188777686579, 3.63356016971806899270931581536, 4.72121089856995631565104750498, 5.10942836408317068989441894151, 6.23557045964603883301190964202, 6.50631012254476021640822163523, 7.47818031133602201288462997171, 8.181767095037083665039383790276, 9.141835712938481320721142610050, 9.240843808415960017862893798652, 10.18487672862619229966687375147, 11.08638761826484467966094767741, 11.69793327349586923753835272384, 12.32977298346883012399583422873, 12.97924981677093867485116166361, 13.56982560306490774743633173202

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 $\F_{q}$

\(L(19)\)	\(\approx\)	\(3.654604134\)
\(L(\frac12)\)	\(\approx\)	\(3.654604134\)
\(L(\frac{39}{2})\)		not available
\(L(1)\)		not available

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