LMFDB - L-function 4-1216e2-1.1-c0e2-0-0

Dirichlet series

L(s) = 1

− 2·9^-s + 4·17^-s + 2·25^-s − 2·49^-s − 4·73^-s + 3·81^-s − 2·121^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s − 8·153^-s + 157^-s + 163^-s + 167^-s − 2·169^-s + 173^-s + 179^-s + 181^-s + 191^-s + 193^-s + 197^-s + 199^-s + 211^-s + 223^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$1478656$ = $2^{12} \cdot 19^{2}$
Sign:	$1$
Analytic conductor:	$0.368282$
Root analytic conductor:	$0.779014$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 1478656,\ (\ :0, 0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.009778879$
$L(\frac12)$	$\approx$	$1.009778879$
$L(1)$		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 $
	19	$C_2$	$ 1 + T^{2} $
good	3	$C_2$	$ ( 1 + T^{2} )^{2} $
	5	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	7	$C_2$	$ ( 1 + T^{2} )^{2} $
	11	$C_2$	$ ( 1 + T^{2} )^{2} $
	13	$C_2$	$ ( 1 + T^{2} )^{2} $
	17	$C_1$	$ ( 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	$C_2$	$ ( 1 + T^{2} )^{2} $
	53	$C_2$	$ ( 1 + T^{2} )^{2} $
	59	$C_2$	$ ( 1 + T^{2} )^{2} $
	61	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	67	$C_2$	$ ( 1 + T^{2} )^{2} $
	71	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	73	$C_1$	$ ( 1 + T )^{4} $
	79	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	83	$C_2$	$ ( 1 + T^{2} )^{2} $
	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

−10.03858683488733557500576328186, −9.788327927997988828700733405486, −9.336471726877595065706323718001, −8.846577655230746483293077608280, −8.332150890540909515735825309940, −8.318244500592858537973108643467, −7.60113094711622986676220564521, −7.51386320559291969345131698271, −6.83392596893603686582455254946, −6.25276101971310853271943309746, −5.74416293752405322785256604644, −5.69307302927373193768143545120, −5.06060394405552271932057231691, −4.84572381246513044076514900335, −3.90882605367159703739915161423, −3.27243948113723452794469111396, −2.98582072277298909962376996114, −2.86225146813457197621463621780, −1.62813312193582553836094851059, −0.973135725108569989388929430460, 0.973135725108569989388929430460, 1.62813312193582553836094851059, 2.86225146813457197621463621780, 2.98582072277298909962376996114, 3.27243948113723452794469111396, 3.90882605367159703739915161423, 4.84572381246513044076514900335, 5.06060394405552271932057231691, 5.69307302927373193768143545120, 5.74416293752405322785256604644, 6.25276101971310853271943309746, 6.83392596893603686582455254946, 7.51386320559291969345131698271, 7.60113094711622986676220564521, 8.318244500592858537973108643467, 8.332150890540909515735825309940, 8.846577655230746483293077608280, 9.336471726877595065706323718001, 9.788327927997988828700733405486, 10.03858683488733557500576328186

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(1.009778879\)
\(L(\frac12)\)	\(\approx\)	\(1.009778879\)
\(L(1)\)		not available
\(L(1)\)		not available

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