LMFDB - L-function 4-475e2-1.1-c0e2-0-0

Dirichlet series

L(s) = 1

− 16^-s + 2·19^-s − 2·49^-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 + 239^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$225625$ = $5^{4} \cdot 19^{2}$
Sign:	$1$
Analytic conductor:	$0.0561954$
Root analytic conductor:	$0.486883$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 225625,\ (\ :0, 0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.7410178968$
$L(\frac12)$	$\approx$	$0.7410178968$
$L(1)$		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	5		$ 1 $
	19	$C_1$	$ ( 1 - T )^{2} $
good	2	$C_2^2$	$ 1 + T^{4} $
	3	$C_2^2$	$ 1 + T^{4} $
	7	$C_2$	$ ( 1 + T^{2} )^{2} $
	11	$C_2$	$ ( 1 + T^{2} )^{2} $
	13	$C_2^2$	$ 1 + T^{4} $
	17	$C_2$	$ ( 1 + T^{2} )^{2} $
	23	$C_2$	$ ( 1 + T^{2} )^{2} $
	29	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	31	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	37	$C_2^2$	$ 1 + T^{4} $
	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	$C_2^2$	$ 1 + T^{4} $
	59	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	61	$C_2$	$ ( 1 + T^{2} )^{2} $
	67	$C_2^2$	$ 1 + T^{4} $
	71	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	73	$C_2$	$ ( 1 + T^{2} )^{2} $
	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	$C_2^2$	$ 1 + T^{4} $
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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

−11.37990582308993058538540547590, −11.17766425103694944005914721705, −10.57623719591102067851626240796, −9.975729415498283986647053493887, −9.718099286941911623165117393099, −9.217191113077171857790777816209, −8.920466863220392946068575919118, −8.111811888096179358965101859191, −7.994480307557649378855852847443, −7.17561271918665208873725262681, −7.03731933014322027177403202566, −6.38299027142892247989561511415, −5.81023703002536076086294186153, −5.30219161385696357359246288479, −4.77822955115321017582691481289, −4.29637028922848141649342123691, −3.44325559732078420873714628887, −3.05790241124699620181144792259, −2.22458540386151061200000898513, −1.31743840664703523636206152464, 1.31743840664703523636206152464, 2.22458540386151061200000898513, 3.05790241124699620181144792259, 3.44325559732078420873714628887, 4.29637028922848141649342123691, 4.77822955115321017582691481289, 5.30219161385696357359246288479, 5.81023703002536076086294186153, 6.38299027142892247989561511415, 7.03731933014322027177403202566, 7.17561271918665208873725262681, 7.994480307557649378855852847443, 8.111811888096179358965101859191, 8.920466863220392946068575919118, 9.217191113077171857790777816209, 9.718099286941911623165117393099, 9.975729415498283986647053493887, 10.57623719591102067851626240796, 11.17766425103694944005914721705, 11.37990582308993058538540547590

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(0.7410178968\)
\(L(\frac12)\)	\(\approx\)	\(0.7410178968\)
\(L(1)\)		not available
\(L(1)\)		not available

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