LMFDB - L-function 4-51e2-1.1-c1e2-0-0

Dirichlet series

L(s) = 1

− 4·2^-s + 8·4^-s − 8·8^-s − 9^-s − 2·13^-s − 4·16^-s + 8·17^-s + 4·18^-s + 10·19^-s + 25^-s + 8·26^-s + 32·32^-s − 32·34^-s − 8·36^-s − 40·38^-s − 2·43^-s − 4·47^-s + 10·49^-s − 4·50^-s − 16·52^-s − 12·53^-s − 64·64^-s − 24·67^-s + 64·68^-s + 8·72^-s + 80·76^-s + 81^-s + ⋯

L(s) = 1

− 2.82·2^-s + 4·4^-s − 2.82·8^-s − 1/3·9^-s − 0.554·13^-s − 16^-s + 1.94·17^-s + 0.942·18^-s + 2.29·19^-s + 1/5·25^-s + 1.56·26^-s + 5.65·32^-s − 5.48·34^-s − 4/3·36^-s − 6.48·38^-s − 0.304·43^-s − 0.583·47^-s + 10/7·49^-s − 0.565·50^-s − 2.21·52^-s − 1.64·53^-s − 8·64^-s − 2.93·67^-s + 7.76·68^-s + 0.942·72^-s + 9.17·76^-s + 1/9·81^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$2601$ = $3^{2} \cdot 17^{2}$
Sign:	$1$
Analytic conductor:	$0.165842$
Root analytic conductor:	$0.638151$
Motivic weight:	$1$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 2601,\ (\ :1/2, 1/2),\ 1)$

Particular Values

$L(1)$	$\approx$	$0.1961016355$
$L(\frac12)$	$\approx$	$0.1961016355$
$L(\frac{3}{2})$		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	3	$C_2$	$ 1 + T^{2} $
	17	$C_2$	$ 1 - 8 T + p T^{2} $
good	2	$C_2$	$ ( 1 + p T + p T^{2} )^{2} $
	5	$C_2^2$	$ 1 - T^{2} + p^{2} T^{4} $
	7	$C_2^2$	$ 1 - 10 T^{2} + p^{2} T^{4} $
	11	$C_2^2$	$ 1 + 3 T^{2} + p^{2} T^{4} $
	13	$C_2$	$ ( 1 + T + p T^{2} )^{2} $
	19	$C_2$	$ ( 1 - 5 T + p T^{2} )^{2} $
	23	$C_2^2$	$ 1 - 45 T^{2} + p^{2} T^{4} $
	29	$C_2^2$	$ 1 - 22 T^{2} + p^{2} T^{4} $
	31	$C_2^2$	$ 1 + 38 T^{2} + p^{2} T^{4} $
	37	$C_2$	$ ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) $
	41	$C_2^2$	$ 1 - 57 T^{2} + p^{2} T^{4} $
	43	$C_2$	$ ( 1 + T + p T^{2} )^{2} $
	47	$C_2$	$ ( 1 + 2 T + p T^{2} )^{2} $
	53	$C_2$	$ ( 1 + 6 T + p T^{2} )^{2} $
	59	$C_2$	$ ( 1 + p T^{2} )^{2} $
	61	$C_2$	$ ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) $
	67	$C_2$	$ ( 1 + 12 T + p T^{2} )^{2} $
	71	$C_2$	$ ( 1 - p T^{2} )^{2} $
	73	$C_2$	$ ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) $
	79	$C_2^2$	$ 1 - 142 T^{2} + p^{2} T^{4} $
	83	$C_2$	$ ( 1 + 6 T + p T^{2} )^{2} $
	89	$C_2$	$ ( 1 + 10 T + p T^{2} )^{2} $
	97	$C_2$	$ ( 1 - 18 T + p T^{2} )( 1 + 18 T + p 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

−16.04530204198886973991244594231, −15.84887845474667030444703591069, −14.90119948851702670542838259770, −14.05191685294401356150237231827, −13.85291752164158867883469039786, −12.81367126406223898632063433193, −11.83048592172730241922760180118, −11.65908178179317882414980154939, −10.79893021295959792211210909271, −10.11702010571057693899380612233, −9.853880726604451897125543567790, −9.319847335107657312326197133184, −8.722653162261079204444829518205, −7.981972831200896075642879654744, −7.48519779331006587519403897841, −7.18842213261785091548349858990, −5.86689401723439675748185924682, −4.85479155988738852051002926776, −3.06339199444238961196745332088, −1.30419036797687803621621177953, 1.30419036797687803621621177953, 3.06339199444238961196745332088, 4.85479155988738852051002926776, 5.86689401723439675748185924682, 7.18842213261785091548349858990, 7.48519779331006587519403897841, 7.981972831200896075642879654744, 8.722653162261079204444829518205, 9.319847335107657312326197133184, 9.853880726604451897125543567790, 10.11702010571057693899380612233, 10.79893021295959792211210909271, 11.65908178179317882414980154939, 11.83048592172730241922760180118, 12.81367126406223898632063433193, 13.85291752164158867883469039786, 14.05191685294401356150237231827, 14.90119948851702670542838259770, 15.84887845474667030444703591069, 16.04530204198886973991244594231

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.1961016355\)
\(L(\frac12)\)	\(\approx\)	\(0.1961016355\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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