LMFDB - L-function 4-44e2-1.1-c1e2-0-0

Dirichlet series

L(s) = 1

− 2·2^-s + 2·4^-s + 2·5^-s − 5·9^-s − 4·10^-s + 8·13^-s − 4·16^-s − 4·17^-s + 10·18^-s + 4·20^-s − 7·25^-s − 16·26^-s + 8·32^-s + 8·34^-s − 10·36^-s + 6·37^-s − 16·41^-s − 10·45^-s − 10·49^-s + 14·50^-s + 16·52^-s − 12·53^-s + 24·61^-s − 8·64^-s + 16·65^-s − 8·68^-s + 8·73^-s + ⋯

L(s) = 1

− 1.41·2^-s + 4^-s + 0.894·5^-s − 5/3·9^-s − 1.26·10^-s + 2.21·13^-s − 16^-s − 0.970·17^-s + 2.35·18^-s + 0.894·20^-s − 7/5·25^-s − 3.13·26^-s + 1.41·32^-s + 1.37·34^-s − 5/3·36^-s + 0.986·37^-s − 2.49·41^-s − 1.49·45^-s − 1.42·49^-s + 1.97·50^-s + 2.21·52^-s − 1.64·53^-s + 3.07·61^-s − 64^-s + 1.98·65^-s − 0.970·68^-s + 0.936·73^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$1936$ = $2^{4} \cdot 11^{2}$
Sign:	$1$
Analytic conductor:	$0.123441$
Root analytic conductor:	$0.592740$
Motivic weight:	$1$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 1936,\ (\ :1/2, 1/2),\ 1)$

Particular Values

$L(1)$	$\approx$	$0.3703087246$
$L(\frac12)$	$\approx$	$0.3703087246$
$L(\frac{3}{2})$		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	$C_2$	$ 1 + p T + p T^{2} $
	11	$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 - T + p T^{2} )^{2} $
	7	$C_2$	$ ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) $
	13	$C_2$	$ ( 1 - 4 T + p T^{2} )^{2} $
	17	$C_2$	$ ( 1 + 2 T + p T^{2} )^{2} $
	19	$C_2$	$ ( 1 + p T^{2} )^{2} $
	23	$C_2$	$ ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) $
	29	$C_2$	$ ( 1 + p T^{2} )^{2} $
	31	$C_2$	$ ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) $
	37	$C_2$	$ ( 1 - 3 T + p T^{2} )^{2} $
	41	$C_2$	$ ( 1 + 8 T + p T^{2} )^{2} $
	43	$C_2$	$ ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) $
	47	$C_2$	$ ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) $
	53	$C_2$	$ ( 1 + 6 T + p T^{2} )^{2} $
	59	$C_2$	$ ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) $
	61	$C_2$	$ ( 1 - 12 T + p T^{2} )^{2} $
	67	$C_2$	$ ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) $
	71	$C_2$	$ ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) $
	73	$C_2$	$ ( 1 - 4 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 - 15 T + p T^{2} )^{2} $
	97	$C_2$	$ ( 1 + 7 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

−13.56863905712999451342489435843, −12.95575045127536336185363327794, −11.50606278511501469560040413868, −11.45125861034521058353374083886, −10.88023677523055899886301116177, −10.03550909718107888433464868208, −9.531323384279074079649336150802, −8.603539619290756001226038948684, −8.595888941067365837350463793393, −7.81893826163275369649161903506, −6.36261389471308870138602900888, −6.29323672924194667424631097798, −5.16796004190121525070885092557, −3.57632858228471856718550416679, −2.01168702564605014700712887669, 2.01168702564605014700712887669, 3.57632858228471856718550416679, 5.16796004190121525070885092557, 6.29323672924194667424631097798, 6.36261389471308870138602900888, 7.81893826163275369649161903506, 8.595888941067365837350463793393, 8.603539619290756001226038948684, 9.531323384279074079649336150802, 10.03550909718107888433464868208, 10.88023677523055899886301116177, 11.45125861034521058353374083886, 11.50606278511501469560040413868, 12.95575045127536336185363327794, 13.56863905712999451342489435843

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.3703087246\)
\(L(\frac12)\)	\(\approx\)	\(0.3703087246\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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