LMFDB - L-function 4-183e2-1.1-c0e2-0-0

Dirichlet series

L(s) = 1

− 2·3^-s + 3·9^-s − 16^-s + 2·25^-s − 4·27^-s + 2·48^-s + 2·49^-s − 2·61^-s − 4·73^-s − 4·75^-s + 5·81^-s − 4·109^-s + 127^-s + 131^-s + 137^-s + 139^-s − 3·144^-s − 4·147^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s − 2·169^-s + 173^-s + 179^-s + 181^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$33489$ = $3^{2} \cdot 61^{2}$
Sign:	$1$
Analytic conductor:	$0.00834096$
Root analytic conductor:	$0.302206$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 33489,\ (\ :0, 0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.2753596510$
$L(\frac12)$	$\approx$	$0.2753596510$
$L(1)$		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	$C_1$	$ ( 1 + T )^{2} $
	61	$C_1$	$ ( 1 + T )^{2} $
good	2	$C_2^2$	$ 1 + T^{4} $
	5	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	7	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	11	$C_2^2$	$ 1 + T^{4} $
	13	$C_2$	$ ( 1 + T^{2} )^{2} $
	17	$C_2^2$	$ 1 + T^{4} $
	19	$C_2$	$ ( 1 + T^{2} )^{2} $
	23	$C_2^2$	$ 1 + T^{4} $
	29	$C_2^2$	$ 1 + T^{4} $
	31	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	37	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	41	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	43	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	47	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	53	$C_2^2$	$ 1 + T^{4} $
	59	$C_2^2$	$ 1 + T^{4} $
	67	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	71	$C_2^2$	$ 1 + T^{4} $
	73	$C_1$	$ ( 1 + T )^{4} $
	79	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	83	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	89	$C_2^2$	$ 1 + T^{4} $
	97	$C_2$	$ ( 1 + T^{2} )^{2} $
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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

−13.03631595072602093843039961165, −12.38059608154886510838514855049, −12.11953725416449572094281884343, −11.68114626871187900433581112301, −11.11795954886805531459029265260, −10.70410017445897266800867626914, −10.43925115117384294982861425918, −9.867954567247524654059244183534, −9.073426690220356819787539988198, −8.857296015536400237529541299715, −7.78344977354955596349576291347, −7.18646828509669977662916339740, −6.86277982873809585884880489537, −6.26991446017201404893330014309, −5.71742979439727236139789333984, −5.12579395998211793256364559476, −4.53250682818961780933031255608, −4.09616643505907531739897542113, −2.79385169794189637840328832684, −1.40925262260813712819147565752, 1.40925262260813712819147565752, 2.79385169794189637840328832684, 4.09616643505907531739897542113, 4.53250682818961780933031255608, 5.12579395998211793256364559476, 5.71742979439727236139789333984, 6.26991446017201404893330014309, 6.86277982873809585884880489537, 7.18646828509669977662916339740, 7.78344977354955596349576291347, 8.857296015536400237529541299715, 9.073426690220356819787539988198, 9.867954567247524654059244183534, 10.43925115117384294982861425918, 10.70410017445897266800867626914, 11.11795954886805531459029265260, 11.68114626871187900433581112301, 12.11953725416449572094281884343, 12.38059608154886510838514855049, 13.03631595072602093843039961165

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(0.2753596510\)
\(L(\frac12)\)	\(\approx\)	\(0.2753596510\)
\(L(1)\)		not available
\(L(1)\)		not available

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