LMFDB - L-function 4-896e2-1.1-c0e2-0-0

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 802816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$802816$ = $2^{14} \cdot 7^{2}$
Sign:	$1$
Analytic conductor:	$0.199954$
Root analytic conductor:	$0.668701$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 802816,\ (\ :0, 0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.6803471836$
$L(\frac12)$	$\approx$	$0.6803471836$
$L(1)$		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_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{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_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	31	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	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_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	59	$C_2^2$	$ 1 + T^{4} $
	61	$C_2^2$	$ 1 + T^{4} $
	67	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{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

−10.60897733640590946990167281856, −9.905047453160838557555947178868, −9.627348474367187936248560614553, −9.527308881005127497001650019730, −8.874171836849939427887581816672, −8.591675587339170309325299746213, −7.918725979961247470488906846370, −7.61559460902891377258952431573, −6.88928402697298387294447091967, −6.77565619617616596172050781987, −6.15758523147932605603450862004, −6.03976651127089508104081777173, −5.14686233740989740911193548778, −5.01908424709103718454780659728, −3.99104710818803354934324698318, −3.73845839731350988791793931379, −3.26737214085796217190633366410, −2.59903864581965609251739485447, −2.11120570570552100753675427989, −0.826826663636152418274709147478, 0.826826663636152418274709147478, 2.11120570570552100753675427989, 2.59903864581965609251739485447, 3.26737214085796217190633366410, 3.73845839731350988791793931379, 3.99104710818803354934324698318, 5.01908424709103718454780659728, 5.14686233740989740911193548778, 6.03976651127089508104081777173, 6.15758523147932605603450862004, 6.77565619617616596172050781987, 6.88928402697298387294447091967, 7.61559460902891377258952431573, 7.918725979961247470488906846370, 8.591675587339170309325299746213, 8.874171836849939427887581816672, 9.527308881005127497001650019730, 9.627348474367187936248560614553, 9.905047453160838557555947178868, 10.60897733640590946990167281856

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(0.6803471836\)
\(L(\frac12)\)	\(\approx\)	\(0.6803471836\)
\(L(1)\)		not available
\(L(1)\)		not available

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