LMFDB - L-function 4-464e2-1.1-c0e2-0-0

Dirichlet series

L(s) = 1

− 2·5^-s + 9^-s + 2·13^-s + 25^-s − 2·29^-s − 2·45^-s + 2·49^-s + 2·53^-s − 4·65^-s − 2·109^-s + 2·117^-s + 121^-s + 2·125^-s + 127^-s + 131^-s + 137^-s + 139^-s + 4·145^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s + 169^-s + 173^-s + 179^-s + 181^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$215296$ = $2^{8} \cdot 29^{2}$
Sign:	$1$
Analytic conductor:	$0.0536228$
Root analytic conductor:	$0.481213$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 215296,\ (\ :0, 0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.5848836986$
$L(\frac12)$	$\approx$	$0.5848836986$
$L(1)$		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	2		$ 1 $
	29	$C_1$	$ ( 1 + T )^{2} $
good	3	$C_2^2$	$ 1 - T^{2} + T^{4} $
	5	$C_2$	$ ( 1 + T + T^{2} )^{2} $
	7	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	11	$C_2^2$	$ 1 - T^{2} + T^{4} $
	13	$C_2$	$ ( 1 - T + T^{2} )^{2} $
	17	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	19	$C_2$	$ ( 1 + T^{2} )^{2} $
	23	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	31	$C_2^2$	$ 1 - T^{2} + T^{4} $
	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	$C_2^2$	$ 1 - T^{2} + T^{4} $
	47	$C_2^2$	$ 1 - T^{2} + T^{4} $
	53	$C_2$	$ ( 1 - T + T^{2} )^{2} $
	59	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	61	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	67	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	71	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	73	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	79	$C_2^2$	$ 1 - T^{2} + T^{4} $
	83	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	89	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	97	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{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

−11.50957380338186536713983981303, −11.08479069301562213773353177469, −10.72704417583730323681044513877, −10.32737082283637493454871142287, −9.650694627785527671094315416091, −9.195609838074158071643385835896, −8.540536735855033554834355852081, −8.444318973069423921874689827219, −7.78531542646813122337448517613, −7.32060454705419037106658802999, −7.20611953289142977531466085623, −6.47450709545356054752913585179, −5.79069064562742616266742104914, −5.45373562200071644882452719981, −4.44374120952213916964669888054, −4.02218389778829586504979194803, −3.80602410384294910190198405510, −3.34523480294010391715411577184, −2.16512437619000855345257718597, −1.14404647117869396583334397581, 1.14404647117869396583334397581, 2.16512437619000855345257718597, 3.34523480294010391715411577184, 3.80602410384294910190198405510, 4.02218389778829586504979194803, 4.44374120952213916964669888054, 5.45373562200071644882452719981, 5.79069064562742616266742104914, 6.47450709545356054752913585179, 7.20611953289142977531466085623, 7.32060454705419037106658802999, 7.78531542646813122337448517613, 8.444318973069423921874689827219, 8.540536735855033554834355852081, 9.195609838074158071643385835896, 9.650694627785527671094315416091, 10.32737082283637493454871142287, 10.72704417583730323681044513877, 11.08479069301562213773353177469, 11.50957380338186536713983981303

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(0.5848836986\)
\(L(\frac12)\)	\(\approx\)	\(0.5848836986\)
\(L(1)\)		not available
\(L(1)\)		not available

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