LMFDB - L-function 4-1792e2-1.1-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

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

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

$L(\frac{1}{2})$	$\approx$	$0.7484047182$
$L(\frac12)$	$\approx$	$0.7484047182$
$L(1)$		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

−9.735451752035746112398946584145, −9.433913439247296679071091604554, −8.874239992228947871295029907154, −8.607026814142739218987755268580, −8.134716525764504755790280969845, −7.68073789616821305850212967561, −7.13174827753211292367713663677, −6.79743285310743289059958710336, −6.67975245207948186020533603510, −5.91293476356486502448103500445, −5.87634861256748669143952084556, −5.30861785228837432171374603994, −4.77179641024740516438809012266, −4.00475312761389532604162435734, −3.96565929565125957551253785738, −3.22204258923318931906077651284, −2.91916260714663413256731236995, −2.42026367985485045517483883468, −1.65987128712138572885536344919, −0.65117120693111178818101502296, 0.65117120693111178818101502296, 1.65987128712138572885536344919, 2.42026367985485045517483883468, 2.91916260714663413256731236995, 3.22204258923318931906077651284, 3.96565929565125957551253785738, 4.00475312761389532604162435734, 4.77179641024740516438809012266, 5.30861785228837432171374603994, 5.87634861256748669143952084556, 5.91293476356486502448103500445, 6.67975245207948186020533603510, 6.79743285310743289059958710336, 7.13174827753211292367713663677, 7.68073789616821305850212967561, 8.134716525764504755790280969845, 8.607026814142739218987755268580, 8.874239992228947871295029907154, 9.433913439247296679071091604554, 9.735451752035746112398946584145

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(0.7484047182\)
\(L(\frac12)\)	\(\approx\)	\(0.7484047182\)
\(L(1)\)		not available
\(L(1)\)		not available

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