LMFDB - L-function 4-1536e2-1.1-c0e2-0-1

Dirichlet series

L(s) = 1

+ 2·3^-s + 3·9^-s + 4·27^-s − 4·59^-s + 5·81^-s − 4·97^-s − 4·107^-s − 2·121^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s + 2·169^-s + 173^-s − 8·177^-s + 179^-s + 181^-s + 191^-s + 193^-s + 197^-s + 199^-s + 211^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$2359296$ = $2^{18} \cdot 3^{2}$
Sign:	$1$
Analytic conductor:	$0.587620$
Root analytic conductor:	$0.875536$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 2359296,\ (\ :0, 0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.485087654$
$L(\frac12)$	$\approx$	$2.485087654$
$L(1)$		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 $
	3	$C_1$	$ ( 1 - T )^{2} $
good	5	$C_2^2$	$ 1 + T^{4} $
	7	$C_2^2$	$ 1 + T^{4} $
	11	$C_2$	$ ( 1 + T^{2} )^{2} $
	13	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	17	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	19	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	23	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	29	$C_2^2$	$ 1 + T^{4} $
	31	$C_2^2$	$ 1 + 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_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	$C_1$	$ ( 1 + T )^{4} $
	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	$C_2$	$ ( 1 + T^{2} )^{2} $
	79	$C_2^2$	$ 1 + T^{4} $
	83	$C_2$	$ ( 1 + T^{2} )^{2} $
	89	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	97	$C_1$	$ ( 1 + 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

−9.550921394166160847716498469244, −9.413896887718742289830315966361, −9.176270549042840228222865053889, −8.588272231777503726473238707047, −8.176367708210517176732371886715, −8.059010671981256490513256058603, −7.57409758163112340126900930688, −7.16165400789348676851828029969, −6.77955257694320601512959527309, −6.39924665153750277548077557574, −5.77154670317581302555597347678, −5.23075908150233779266154236587, −4.53881449803914703632908118021, −4.39498908294283308545861657345, −3.78791307046467432961624445028, −3.36678728065054566479336907641, −2.68649677726142148424106723441, −2.67568195238321346961721808635, −1.63855927950080146808216236246, −1.42681498803969396025646528492, 1.42681498803969396025646528492, 1.63855927950080146808216236246, 2.67568195238321346961721808635, 2.68649677726142148424106723441, 3.36678728065054566479336907641, 3.78791307046467432961624445028, 4.39498908294283308545861657345, 4.53881449803914703632908118021, 5.23075908150233779266154236587, 5.77154670317581302555597347678, 6.39924665153750277548077557574, 6.77955257694320601512959527309, 7.16165400789348676851828029969, 7.57409758163112340126900930688, 8.059010671981256490513256058603, 8.176367708210517176732371886715, 8.588272231777503726473238707047, 9.176270549042840228222865053889, 9.413896887718742289830315966361, 9.550921394166160847716498469244

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(2.485087654\)
\(L(\frac12)\)	\(\approx\)	\(2.485087654\)
\(L(1)\)		not available
\(L(1)\)		not available

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