LMFDB - L-function 4-3536e2-1.1-c0e2-0-9

Dirichlet series

L(s) = 1

− 2·13^-s + 2·17^-s + 4·19^-s + 4·47^-s + 2·49^-s − 4·59^-s − 81^-s + 2·121^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s + 3·169^-s + 173^-s + 179^-s + 181^-s + 191^-s + 193^-s + 197^-s + 199^-s + 211^-s − 4·221^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12503296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$12503296$ = $2^{8} \cdot 13^{2} \cdot 17^{2}$
Sign:	$1$
Analytic conductor:	$3.11414$
Root analytic conductor:	$1.32841$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 12503296,\ (\ :0, 0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.719149484$
$L(\frac12)$	$\approx$	$1.719149484$
$L(1)$		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 $
	13	$C_1$	$ ( 1 + T )^{2} $
	17	$C_1$	$ ( 1 - T )^{2} $
good	3	$C_2^2$	$ 1 + T^{4} $
	5	$C_2^2$	$ 1 + T^{4} $
	7	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	11	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	19	$C_1$	$ ( 1 - T )^{4} $
	23	$C_2^2$	$ 1 + T^{4} $
	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	$C_2^2$	$ 1 + T^{4} $
	43	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	47	$C_1$	$ ( 1 - T )^{4} $
	53	$C_2$	$ ( 1 + T^{2} )^{2} $
	59	$C_1$	$ ( 1 + T )^{4} $
	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	$C_2^2$	$ 1 + T^{4} $
	79	$C_2^2$	$ 1 + T^{4} $
	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

−9.063309784625067404324276022961, −8.675637823845772098176674196827, −7.84328873052770445550189516704, −7.68630064108811194885570007849, −7.52091772017344907568165291298, −7.22106035986045796352471878956, −7.06845467152089405858334678591, −6.20206738588399614554337900135, −5.62221771801975913554735867536, −5.57444764831798250192042890835, −5.38416862297676243729786241098, −4.75915458462822757358046538981, −4.44869582788402907342451705038, −3.85792070298300776592806113612, −3.17060782278750909088483234794, −3.13880997315856967261120757532, −2.67109934305354037353681252106, −2.05902944977426339903672140167, −1.11845333704597679631350297651, −0.983172168132892547495627375627, 0.983172168132892547495627375627, 1.11845333704597679631350297651, 2.05902944977426339903672140167, 2.67109934305354037353681252106, 3.13880997315856967261120757532, 3.17060782278750909088483234794, 3.85792070298300776592806113612, 4.44869582788402907342451705038, 4.75915458462822757358046538981, 5.38416862297676243729786241098, 5.57444764831798250192042890835, 5.62221771801975913554735867536, 6.20206738588399614554337900135, 7.06845467152089405858334678591, 7.22106035986045796352471878956, 7.52091772017344907568165291298, 7.68630064108811194885570007849, 7.84328873052770445550189516704, 8.675637823845772098176674196827, 9.063309784625067404324276022961

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(1.719149484\)
\(L(\frac12)\)	\(\approx\)	\(1.719149484\)
\(L(1)\)		not available
\(L(1)\)		not available

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