LMFDB - L-function 4-896e2-1.1-c0e2-0-3

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

$L(\frac{1}{2})$	$\approx$	$1.194020124$
$L(\frac12)$	$\approx$	$1.194020124$
$L(1)$		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

−10.44660375672113490465912434070, −10.14746318908492613647259307666, −9.914221623344445245420765247528, −8.969697968756809860395976038882, −8.797568559389099666788830059643, −8.596667266846424619982833358831, −7.955671642138657226801464781656, −7.51603439764461457949066151747, −7.38088076323783862002257168116, −6.78870191631647409778969357441, −6.10923970358682529950724072724, −5.59548830859865741659255979477, −5.40223632646380237037549214418, −4.53341792128439175629482969925, −4.51621428394311017800297436680, −3.95256151711235072160277779437, −3.04862959745114902768635689818, −2.57535299837503157275197784454, −1.69662029269896584039931522169, −1.35654336297365492313584603706, 1.35654336297365492313584603706, 1.69662029269896584039931522169, 2.57535299837503157275197784454, 3.04862959745114902768635689818, 3.95256151711235072160277779437, 4.51621428394311017800297436680, 4.53341792128439175629482969925, 5.40223632646380237037549214418, 5.59548830859865741659255979477, 6.10923970358682529950724072724, 6.78870191631647409778969357441, 7.38088076323783862002257168116, 7.51603439764461457949066151747, 7.955671642138657226801464781656, 8.596667266846424619982833358831, 8.797568559389099666788830059643, 8.969697968756809860395976038882, 9.914221623344445245420765247528, 10.14746318908492613647259307666, 10.44660375672113490465912434070

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(1.194020124\)
\(L(\frac12)\)	\(\approx\)	\(1.194020124\)
\(L(1)\)		not available
\(L(1)\)		not available

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