LMFDB - L-function 4-11552-1.1-c1e2-0-0

Dirichlet series

L(s) = 1

− 2^-s + 4^-s − 8^-s − 5·9^-s + 10·13^-s + 16^-s + 6·17^-s + 5·18^-s − 10·25^-s − 10·26^-s + 18·29^-s − 32^-s − 6·34^-s − 5·36^-s + 4·37^-s − 13·49^-s + 10·50^-s + 10·52^-s − 6·53^-s − 18·58^-s − 20·61^-s + 64^-s + 6·68^-s + 5·72^-s − 14·73^-s − 4·74^-s + 16·81^-s + ⋯

L(s) = 1

− 0.707·2^-s + 1/2·4^-s − 0.353·8^-s − 5/3·9^-s + 2.77·13^-s + 1/4·16^-s + 1.45·17^-s + 1.17·18^-s − 2·25^-s − 1.96·26^-s + 3.34·29^-s − 0.176·32^-s − 1.02·34^-s − 5/6·36^-s + 0.657·37^-s − 1.85·49^-s + 1.41·50^-s + 1.38·52^-s − 0.824·53^-s − 2.36·58^-s − 2.56·61^-s + 1/8·64^-s + 0.727·68^-s + 0.589·72^-s − 1.63·73^-s − 0.464·74^-s + 16/9·81^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11552 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 11552 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$11552$ = $2^{5} \cdot 19^{2}$
Sign:	$1$
Analytic conductor:	$0.736565$
Root analytic conductor:	$0.926409$
Motivic weight:	$1$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 11552,\ (\ :1/2, 1/2),\ 1)$

Particular Values

$L(1)$	$\approx$	$0.7579089332$
$L(\frac12)$	$\approx$	$0.7579089332$
$L(\frac{3}{2})$		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	$C_1$	$ 1 + T $
	19	$C_1$$\times$$C_1$	$ ( 1 - T )( 1 + T ) $
good	3	$C_2$	$ ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) $
	5	$C_2$	$ ( 1 + p T^{2} )^{2} $
	7	$C_2$	$ ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) $
	11	$C_2$	$ ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) $
	13	$C_2$	$ ( 1 - 5 T + p T^{2} )^{2} $
	17	$C_2$	$ ( 1 - 3 T + p T^{2} )^{2} $
	23	$C_2$	$ ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) $
	29	$C_2$	$ ( 1 - 9 T + p T^{2} )^{2} $
	31	$C_2$	$ ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) $
	37	$C_2$	$ ( 1 - 2 T + p T^{2} )^{2} $
	41	$C_2$	$ ( 1 + p T^{2} )^{2} $
	43	$C_2$	$ ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) $
	47	$C_2$	$ ( 1 + p T^{2} )^{2} $
	53	$C_2$	$ ( 1 + 3 T + p T^{2} )^{2} $
	59	$C_2$	$ ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) $
	61	$C_2$	$ ( 1 + 10 T + p T^{2} )^{2} $
	67	$C_2$	$ ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) $
	71	$C_2$	$ ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) $
	73	$C_2$	$ ( 1 + 7 T + p T^{2} )^{2} $
	79	$C_2$	$ ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) $
	83	$C_2$	$ ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) $
	89	$C_2$	$ ( 1 + 12 T + p T^{2} )^{2} $
	97	$C_2$	$ ( 1 + 10 T + p T^{2} )^{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

−11.24807629195878921565630077767, −11.01515102281886456905362701904, −10.09897479975324709020702728875, −9.913028354445878842381293206945, −8.798645403370168187312975017332, −8.684413720393242419277779815238, −8.059349923675553885391410616419, −7.73591843742988285704805588989, −6.44167245371229223240488859667, −6.04366496258479301724306565185, −5.81027585652647529822118508915, −4.58652942277425117027930400256, −3.41440775482142096356524512987, −2.99432139105097413830284496966, −1.34050914666900867420115229318, 1.34050914666900867420115229318, 2.99432139105097413830284496966, 3.41440775482142096356524512987, 4.58652942277425117027930400256, 5.81027585652647529822118508915, 6.04366496258479301724306565185, 6.44167245371229223240488859667, 7.73591843742988285704805588989, 8.059349923675553885391410616419, 8.684413720393242419277779815238, 8.798645403370168187312975017332, 9.913028354445878842381293206945, 10.09897479975324709020702728875, 11.01515102281886456905362701904, 11.24807629195878921565630077767

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7579089332\)
\(L(\frac12)\)	\(\approx\)	\(0.7579089332\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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