LMFDB - L-function 4-799e2-1.1-c0e2-0-1

Dirichlet series

L(s) = 1

− 2·4^-s + 2·9^-s + 3·16^-s + 2·17^-s − 4·36^-s + 2·47^-s + 2·49^-s − 4·53^-s − 4·64^-s − 4·68^-s + 3·81^-s − 4·83^-s − 4·89^-s + 127^-s + 131^-s + 137^-s + 139^-s + 6·144^-s + 149^-s + 151^-s + 4·153^-s + 157^-s + 163^-s + 167^-s + 2·169^-s + 173^-s + 179^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 638401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$638401$ = $17^{2} \cdot 47^{2}$
Sign:	$1$
Analytic conductor:	$0.159003$
Root analytic conductor:	$0.631468$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 638401,\ (\ :0, 0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.7261492744$
$L(\frac12)$	$\approx$	$0.7261492744$
$L(1)$		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	17	$C_1$	$ ( 1 - T )^{2} $
	47	$C_1$	$ ( 1 - T )^{2} $
good	2	$C_2$	$ ( 1 + T^{2} )^{2} $
	3	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	5	$C_2^2$	$ 1 + T^{4} $
	7	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	11	$C_2^2$	$ 1 + T^{4} $
	13	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	19	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	23	$C_2^2$	$ 1 + T^{4} $
	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	$C_2^2$	$ 1 + T^{4} $
	43	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	53	$C_1$	$ ( 1 + T )^{4} $
	59	$C_2$	$ ( 1 + T^{2} )^{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	$C_2^2$	$ 1 + T^{4} $
	79	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	83	$C_1$	$ ( 1 + T )^{4} $
	89	$C_1$	$ ( 1 + T )^{4} $
	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

−10.33348361815926683397981824937, −10.20096756840628288540195937784, −9.746689803232361517317193832081, −9.525512684286570793824743923626, −9.153165933436031485356074601632, −8.601506233735390959462843882198, −8.063400185100257425644525510662, −7.87659228110464969574719270538, −7.17969365198483845132119138473, −7.15680822313243672563890265455, −6.16401959531337729925372970235, −5.60388819097852932686624349106, −5.42920296113198987194662399730, −4.70967097461339598712791931569, −4.23653969168912101210015572507, −4.13220800955407124516127928695, −3.42870379421018278691341708730, −2.89506002251114788168038342748, −1.48972277724596277388070807012, −1.14367168884042813850767301987, 1.14367168884042813850767301987, 1.48972277724596277388070807012, 2.89506002251114788168038342748, 3.42870379421018278691341708730, 4.13220800955407124516127928695, 4.23653969168912101210015572507, 4.70967097461339598712791931569, 5.42920296113198987194662399730, 5.60388819097852932686624349106, 6.16401959531337729925372970235, 7.15680822313243672563890265455, 7.17969365198483845132119138473, 7.87659228110464969574719270538, 8.063400185100257425644525510662, 8.601506233735390959462843882198, 9.153165933436031485356074601632, 9.525512684286570793824743923626, 9.746689803232361517317193832081, 10.20096756840628288540195937784, 10.33348361815926683397981824937

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(0.7261492744\)
\(L(\frac12)\)	\(\approx\)	\(0.7261492744\)
\(L(1)\)		not available
\(L(1)\)		not available

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