LMFDB - L-function 4-1792e2-1.1-c0e2-0-1

Dirichlet series

L(s) = 1

+ 2·7^-s + 3·49^-s − 4·71^-s + 4·79^-s − 81^-s − 2·121^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s + 173^-s + 179^-s + 181^-s + 191^-s + 193^-s + 197^-s + 199^-s + 211^-s + 223^-s + 227^-s + 229^-s + 233^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$3211264$ = $2^{16} \cdot 7^{2}$
Sign:	$1$
Analytic conductor:	$0.799816$
Root analytic conductor:	$0.945687$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 3211264,\ (\ :0, 0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.563689813$
$L(\frac12)$	$\approx$	$1.563689813$
$L(1)$		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 $
	7	$C_1$	$ ( 1 - T )^{2} $
good	3	$C_2^2$	$ 1 + T^{4} $
	5	$C_2^2$	$ 1 + T^{4} $
	11	$C_2$	$ ( 1 + T^{2} )^{2} $
	13	$C_2^2$	$ 1 + T^{4} $
	17	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	19	$C_2^2$	$ 1 + T^{4} $
	23	$C_2$	$ ( 1 + T^{2} )^{2} $
	29	$C_2$	$ ( 1 + T^{2} )^{2} $
	31	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	37	$C_2$	$ ( 1 + T^{2} )^{2} $
	41	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	43	$C_2$	$ ( 1 + T^{2} )^{2} $
	47	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	53	$C_2$	$ ( 1 + T^{2} )^{2} $
	59	$C_2^2$	$ 1 + T^{4} $
	61	$C_2^2$	$ 1 + T^{4} $
	67	$C_2$	$ ( 1 + T^{2} )^{2} $
	71	$C_1$	$ ( 1 + T )^{4} $
	73	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	79	$C_1$	$ ( 1 - T )^{4} $
	83	$C_2^2$	$ 1 + T^{4} $
	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

−9.436501738147963960853144788844, −9.317916696227728669737806947852, −8.828063124266923537357810979182, −8.333072327844836786931180348852, −8.208614030908618842026164717704, −7.75219947748037373603169694053, −7.33932178996036129802160783153, −7.07184758740547440281949013479, −6.51547672919430356488126748274, −5.79700880500421243493978947873, −5.77987139447897251279740835178, −5.10457318126907636583463761981, −4.70527533771882304071814851848, −4.48700696904959416564790014394, −3.91151320164951705671147964477, −3.38682950022807717647016684984, −2.69349438610443807656799239780, −2.16584870428135512906451257331, −1.63431357538949480938595179249, −1.06167667342120008976889295558, 1.06167667342120008976889295558, 1.63431357538949480938595179249, 2.16584870428135512906451257331, 2.69349438610443807656799239780, 3.38682950022807717647016684984, 3.91151320164951705671147964477, 4.48700696904959416564790014394, 4.70527533771882304071814851848, 5.10457318126907636583463761981, 5.77987139447897251279740835178, 5.79700880500421243493978947873, 6.51547672919430356488126748274, 7.07184758740547440281949013479, 7.33932178996036129802160783153, 7.75219947748037373603169694053, 8.208614030908618842026164717704, 8.333072327844836786931180348852, 8.828063124266923537357810979182, 9.317916696227728669737806947852, 9.436501738147963960853144788844

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

\(L(\frac{1}{2})\)	\(\approx\)	\(1.563689813\)
\(L(\frac12)\)	\(\approx\)	\(1.563689813\)
\(L(1)\)		not available
\(L(1)\)		not available

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