LMFDB - L-function 4-3447e2-1.1-c0e2-0-0

Dirichlet series

L(s) = 1

− 2·2^-s − 3^-s + 4^-s + 2·6^-s + 7^-s + 2·8^-s − 12^-s − 2·14^-s − 4·16^-s + 4·17^-s − 2·19^-s − 21^-s + 23^-s − 2·24^-s − 25^-s + 27^-s + 28^-s − 2·29^-s + 31^-s + 2·32^-s − 8·34^-s + 4·38^-s + 2·42^-s + 43^-s − 2·46^-s + 4·48^-s + 49^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11881809 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$11881809$ = $3^{4} \cdot 383^{2}$
Sign:	$1$
Analytic conductor:	$2.95935$
Root analytic conductor:	$1.31159$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 11881809,\ (\ :0, 0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3357753252$
$L(\frac12)$	$\approx$	$0.3357753252$
$L(1)$		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	$C_2$	$ 1 + T + T^{2} $
	383	$C_2$	$ 1 + T + T^{2} $
good	2	$C_2$	$ ( 1 + T + T^{2} )^{2} $
	5	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	7	$C_1$$\times$$C_2$	$ ( 1 - T )^{2}( 1 + T + T^{2} ) $
	11	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	13	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	17	$C_1$	$ ( 1 - T )^{4} $
	19	$C_2$	$ ( 1 + T + T^{2} )^{2} $
	23	$C_1$$\times$$C_2$	$ ( 1 - T )^{2}( 1 + T + T^{2} ) $
	29	$C_2$	$ ( 1 + T + T^{2} )^{2} $
	31	$C_1$$\times$$C_2$	$ ( 1 - T )^{2}( 1 + T + T^{2} ) $
	37	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	41	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	43	$C_1$$\times$$C_2$	$ ( 1 - T )^{2}( 1 + T + T^{2} ) $
	47	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	53	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	59	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	61	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	67	$C_1$$\times$$C_2$	$ ( 1 - T )^{2}( 1 + T + T^{2} ) $
	71	$C_1$	$ ( 1 - T )^{4} $
	73	$C_2$	$ ( 1 + T + T^{2} )^{2} $
	79	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	83	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	89	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	97	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + 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

−8.876141572125379484303826821074, −8.599435511363899286675549423372, −8.179068784508624426509392140388, −7.936623142374043068518553407768, −7.73544085133132194510622876070, −7.29855391955631962510291869618, −7.11293910211326489483195098567, −6.23620953235885083218767242523, −6.12421118243404085339291279868, −5.39686681636427766317560075311, −5.27200701267526622372986351782, −5.01830736826058949748675247897, −4.40756877126365825631447628704, −3.80423744629439419024739642909, −3.75332317368302651949919603780, −2.81652165124919203129595093141, −2.08510886872333285965977350737, −1.63467507950997505719032898373, −1.03315741384425621364575396256, −0.68724975887678589964131551747, 0.68724975887678589964131551747, 1.03315741384425621364575396256, 1.63467507950997505719032898373, 2.08510886872333285965977350737, 2.81652165124919203129595093141, 3.75332317368302651949919603780, 3.80423744629439419024739642909, 4.40756877126365825631447628704, 5.01830736826058949748675247897, 5.27200701267526622372986351782, 5.39686681636427766317560075311, 6.12421118243404085339291279868, 6.23620953235885083218767242523, 7.11293910211326489483195098567, 7.29855391955631962510291869618, 7.73544085133132194510622876070, 7.936623142374043068518553407768, 8.179068784508624426509392140388, 8.599435511363899286675549423372, 8.876141572125379484303826821074

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(0.3357753252\)
\(L(\frac12)\)	\(\approx\)	\(0.3357753252\)
\(L(1)\)		not available
\(L(1)\)		not available

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