LMFDB - L-function 4-2624e2-1.1-c0e2-0-0

Dirichlet series

L(s) = 1

− 2·25^-s + 2·41^-s + 4·61^-s − 81^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s + 2·169^-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 6885376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$6885376$ = $2^{12} \cdot 41^{2}$
Sign:	$1$
Analytic conductor:	$1.71491$
Root analytic conductor:	$1.14435$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 6885376,\ (\ :0, 0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.207253334$
$L(\frac12)$	$\approx$	$1.207253334$
$L(1)$		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 $
	41	$C_1$	$ ( 1 - T )^{2} $
good	3	$C_2^2$	$ 1 + T^{4} $
	5	$C_2$	$ ( 1 + T^{2} )^{2} $
	7	$C_2^2$	$ 1 + T^{4} $
	11	$C_2^2$	$ 1 + T^{4} $
	13	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	17	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	19	$C_2^2$	$ 1 + T^{4} $
	23	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{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	$C_2$	$ ( 1 + T^{2} )^{2} $
	43	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	47	$C_2^2$	$ 1 + T^{4} $
	53	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	59	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	61	$C_1$	$ ( 1 - T )^{4} $
	67	$C_2^2$	$ 1 + T^{4} $
	71	$C_2^2$	$ 1 + T^{4} $
	73	$C_2$	$ ( 1 + T^{2} )^{2} $
	79	$C_2^2$	$ 1 + T^{4} $
	83	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	89	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	97	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + 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

−9.280907958022290083985316068125, −8.861233714949802991324685644457, −8.375614469347444373140465859671, −8.179026272766001121000740546663, −7.74554474997688457206723205875, −7.27330438889949300786059855071, −7.13526760690638206462475322907, −6.48914747921063413796135061090, −6.17861051868916698730733355412, −5.75697920079382410204325177622, −5.38050581067249907645391772889, −5.08480677569838671492988258508, −4.21953390371656761064663233021, −4.21247143803608509772161973167, −3.69504420127389968519596264318, −3.18749595835430474944273979542, −2.43882190169921353977936065846, −2.26655906144415210864048456258, −1.54370962335477847069710564947, −0.73713795966271918933991058572, 0.73713795966271918933991058572, 1.54370962335477847069710564947, 2.26655906144415210864048456258, 2.43882190169921353977936065846, 3.18749595835430474944273979542, 3.69504420127389968519596264318, 4.21247143803608509772161973167, 4.21953390371656761064663233021, 5.08480677569838671492988258508, 5.38050581067249907645391772889, 5.75697920079382410204325177622, 6.17861051868916698730733355412, 6.48914747921063413796135061090, 7.13526760690638206462475322907, 7.27330438889949300786059855071, 7.74554474997688457206723205875, 8.179026272766001121000740546663, 8.375614469347444373140465859671, 8.861233714949802991324685644457, 9.280907958022290083985316068125

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(1.207253334\)
\(L(\frac12)\)	\(\approx\)	\(1.207253334\)
\(L(1)\)		not available
\(L(1)\)		not available

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