LMFDB - L-function 4-3e4-1.1-c33e2-0-0

Dirichlet series

L(s) = 1

+ 1.21e5·2^-s + 6.13e9·4^-s + 1.81e11·5^-s − 6.71e13·7^-s + 7.37e14·8^-s + 2.20e16·10^-s − 1.33e17·11^-s − 2.98e18·13^-s − 8.17e18·14^-s + 8.99e19·16^-s + 7.93e19·17^-s − 1.36e21·19^-s + 1.11e21·20^-s − 1.62e22·22^-s − 2.61e22·23^-s − 1.95e23·25^-s − 3.62e23·26^-s − 4.12e23·28^-s + 1.67e24·29^-s − 6.23e24·31^-s + 1.99e24·32^-s + 9.65e24·34^-s − 1.21e25·35^-s − 1.04e26·37^-s − 1.65e26·38^-s + 1.33e26·40^-s − 2.77e26·41^-s + ⋯

L(s) = 1

+ 1.31·2^-s + 0.714·4^-s + 0.530·5^-s − 0.763·7^-s + 0.926·8^-s + 0.696·10^-s − 0.878·11^-s − 1.24·13^-s − 1.00·14^-s + 1.21·16^-s + 0.395·17^-s − 1.08·19^-s + 0.379·20^-s − 1.15·22^-s − 0.889·23^-s − 1.68·25^-s − 1.63·26^-s − 0.545·28^-s + 1.24·29^-s − 1.54·31^-s + 0.290·32^-s + 0.519·34^-s − 0.405·35^-s − 1.39·37^-s − 1.42·38^-s + 0.491·40^-s − 0.679·41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+33/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$81$ = $3^{4}$
Sign:	$1$
Analytic conductor:	$3854.49$
Root analytic conductor:	$7.87937$
Motivic weight:	$33$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 81,\ (\ :33/2, 33/2),\ 1)$

Particular Values

$L(17)$	$\approx$	$0.02121514055$
$L(\frac12)$	$\approx$	$0.02121514055$
$L(\frac{35}{2})$		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 - 7605 p^{4} T + 2115925 p^{12} T^{2} - 7605 p^{37} T^{3} + p^{66} T^{4} $
	5	$D_{4}$	$ 1 - 1448492292 p^{3} T + 585507855025144558 p^{8} T^{2} - 1448492292 p^{36} T^{3} + p^{66} T^{4} $
	7	$D_{4}$	$ 1 + 9593297152400 p T + $$36\!\cdots\!50$$ p^{4} T^{2} + 9593297152400 p^{34} T^{3} + p^{66} T^{4} $
	11	$D_{4}$	$ 1 + 12170165040174024 p T + $$23\!\cdots\!66$$ p^{2} T^{2} + 12170165040174024 p^{34} T^{3} + p^{66} T^{4} $
	13	$D_{4}$	$ 1 + 229354652173803380 p T + $$61\!\cdots\!50$$ p^{3} T^{2} + 229354652173803380 p^{34} T^{3} + p^{66} T^{4} $
	17	$D_{4}$	$ 1 - 79361149261175525340 T - $$26\!\cdots\!50$$ p T^{2} - 79361149261175525340 p^{33} T^{3} + p^{66} T^{4} $
	19	$D_{4}$	$ 1 + $$13\!\cdots\!00$$ T + $$17\!\cdots\!22$$ p T^{2} + $$13\!\cdots\!00$$ p^{33} T^{3} + p^{66} T^{4} $
	23	$D_{4}$	$ 1 + $$26\!\cdots\!40$$ T + $$68\!\cdots\!50$$ p T^{2} + $$26\!\cdots\!40$$ p^{33} T^{3} + p^{66} T^{4} $
	29	$D_{4}$	$ 1 - $$57\!\cdots\!00$$ p T + $$46\!\cdots\!58$$ p^{2} T^{2} - $$57\!\cdots\!00$$ p^{34} T^{3} + p^{66} T^{4} $
	31	$D_{4}$	$ 1 + $$20\!\cdots\!36$$ p T + $$30\!\cdots\!86$$ p^{2} T^{2} + $$20\!\cdots\!36$$ p^{34} T^{3} + p^{66} T^{4} $
	37	$D_{4}$	$ 1 + $$10\!\cdots\!20$$ T + $$13\!\cdots\!50$$ T^{2} + $$10\!\cdots\!20$$ p^{33} T^{3} + p^{66} T^{4} $
	41	$D_{4}$	$ 1 + $$27\!\cdots\!44$$ T + $$22\!\cdots\!26$$ T^{2} + $$27\!\cdots\!44$$ p^{33} T^{3} + p^{66} T^{4} $
	43	$D_{4}$	$ 1 - $$15\!\cdots\!00$$ T + $$12\!\cdots\!50$$ T^{2} - $$15\!\cdots\!00$$ p^{33} T^{3} + p^{66} T^{4} $
	47	$D_{4}$	$ 1 + $$11\!\cdots\!20$$ p T + $$37\!\cdots\!50$$ T^{2} + $$11\!\cdots\!20$$ p^{34} T^{3} + p^{66} T^{4} $
	53	$D_{4}$	$ 1 - $$26\!\cdots\!20$$ T + $$12\!\cdots\!50$$ T^{2} - $$26\!\cdots\!20$$ p^{33} T^{3} + p^{66} T^{4} $
	59	$D_{4}$	$ 1 - $$30\!\cdots\!00$$ T + $$77\!\cdots\!58$$ T^{2} - $$30\!\cdots\!00$$ p^{33} T^{3} + p^{66} T^{4} $
	61	$D_{4}$	$ 1 + $$57\!\cdots\!36$$ T + $$16\!\cdots\!86$$ T^{2} + $$57\!\cdots\!36$$ p^{33} T^{3} + p^{66} T^{4} $
	67	$D_{4}$	$ 1 - $$15\!\cdots\!60$$ T + $$41\!\cdots\!50$$ T^{2} - $$15\!\cdots\!60$$ p^{33} T^{3} + p^{66} T^{4} $
	71	$D_{4}$	$ 1 - $$26\!\cdots\!76$$ T + $$22\!\cdots\!66$$ T^{2} - $$26\!\cdots\!76$$ p^{33} T^{3} + p^{66} T^{4} $
	73	$D_{4}$	$ 1 - $$94\!\cdots\!40$$ T + $$19\!\cdots\!50$$ T^{2} - $$94\!\cdots\!40$$ p^{33} T^{3} + p^{66} T^{4} $
	79	$D_{4}$	$ 1 + $$85\!\cdots\!00$$ T + $$85\!\cdots\!78$$ T^{2} + $$85\!\cdots\!00$$ p^{33} T^{3} + p^{66} T^{4} $
	83	$D_{4}$	$ 1 + $$29\!\cdots\!20$$ T + $$37\!\cdots\!50$$ T^{2} + $$29\!\cdots\!20$$ p^{33} T^{3} + p^{66} T^{4} $
	89	$D_{4}$	$ 1 + $$13\!\cdots\!00$$ T + $$47\!\cdots\!38$$ T^{2} + $$13\!\cdots\!00$$ p^{33} T^{3} + p^{66} T^{4} $
	97	$D_{4}$	$ 1 + $$36\!\cdots\!60$$ T + $$42\!\cdots\!50$$ T^{2} + $$36\!\cdots\!60$$ p^{33} T^{3} + p^{66} 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

−14.08015330685980784913447413397, −13.46912948139148517203353618901, −12.76814587770784757622545255447, −12.54422226576142171013309887400, −11.65882201042686300120446471056, −10.62469884087428772281734725786, −10.07289973263252693515985927038, −9.622016537176720860979209534186, −8.411574983212849430759663194070, −7.66309016926825645684712359703, −6.93953776391012555694629037343, −6.18307145153677745740702837569, −5.26228145783937502490386112774, −5.19962423923673568072910494536, −4.00795900289828917689815351764, −3.77175510616389440508152230795, −2.62532772055263355972238419670, −2.21738605633794544192925859210, −1.46123154365470444616523474221, −0.02658994385999218551493683890, 0.02658994385999218551493683890, 1.46123154365470444616523474221, 2.21738605633794544192925859210, 2.62532772055263355972238419670, 3.77175510616389440508152230795, 4.00795900289828917689815351764, 5.19962423923673568072910494536, 5.26228145783937502490386112774, 6.18307145153677745740702837569, 6.93953776391012555694629037343, 7.66309016926825645684712359703, 8.411574983212849430759663194070, 9.622016537176720860979209534186, 10.07289973263252693515985927038, 10.62469884087428772281734725786, 11.65882201042686300120446471056, 12.54422226576142171013309887400, 12.76814587770784757622545255447, 13.46912948139148517203353618901, 14.08015330685980784913447413397

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 $\F_{q}$

\(L(17)\)	\(\approx\)	\(0.02121514055\)
\(L(\frac12)\)	\(\approx\)	\(0.02121514055\)
\(L(\frac{35}{2})\)		not available
\(L(1)\)		not available

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