LMFDB - L-function 2-4896-8.5-c1-0-4

Dirichlet series

L(s) = 1

− 3.12i·5^-s − 2.86·7^-s − 5.06i·11^-s + 2.04i·13^-s + 17^-s + 6.32i·19^-s − 3.18·23^-s − 4.73·25^-s + 5.45i·29^-s − 6.19·31^-s + 8.95i·35^-s + 1.57i·37^-s + 0.733·41^-s + 5.97i·43^-s − 10.0·47^-s + ⋯

L(s) = 1

− 1.39i·5^-s − 1.08·7^-s − 1.52i·11^-s + 0.566i·13^-s + 0.242·17^-s + 1.45i·19^-s − 0.664·23^-s − 0.947·25^-s + 1.01i·29^-s − 1.11·31^-s + 1.51i·35^-s + 0.258i·37^-s + 0.114·41^-s + 0.910i·43^-s − 1.46·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4896$ = $2^{5} \cdot 3^{2} \cdot 17$
Sign:	$0.409 - 0.912i$
Analytic conductor:	$39.0947$
Root analytic conductor:	$6.25258$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4896} (2449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4896,\ (\ :1/2),\ 0.409 - 0.912i)$

Particular Values

$L(1)$	$\approx$	$0.6285729531$
$L(\frac12)$	$\approx$	$0.6285729531$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	17	$ 1 - T $
good	5	$ 1 + 3.12iT - 5T^{2} $
	7	$ 1 + 2.86T + 7T^{2} $
	11	$ 1 + 5.06iT - 11T^{2} $
	13	$ 1 - 2.04iT - 13T^{2} $
	19	$ 1 - 6.32iT - 19T^{2} $
	23	$ 1 + 3.18T + 23T^{2} $
	29	$ 1 - 5.45iT - 29T^{2} $
	31	$ 1 + 6.19T + 31T^{2} $
	37	$ 1 - 1.57iT - 37T^{2} $
	41	$ 1 - 0.733T + 41T^{2} $
	43	$ 1 - 5.97iT - 43T^{2} $
	47	$ 1 + 10.0T + 47T^{2} $
	53	$ 1 + 10.1iT - 53T^{2} $
	59	$ 1 + 0.268iT - 59T^{2} $
	61	$ 1 + 2.58iT - 61T^{2} $
	67	$ 1 - 0.116iT - 67T^{2} $
	71	$ 1 - 4.49T + 71T^{2} $
	73	$ 1 - 15.6T + 73T^{2} $
	79	$ 1 - 0.815T + 79T^{2} $
	83	$ 1 - 3.81iT - 83T^{2} $
	89	$ 1 - 9.87T + 89T^{2} $
	97	$ 1 + 11.3T + 97T^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−8.265399653404883509932131654722, −8.062837993307020526108296496300, −6.79280703377857803635085766515, −6.13391950480775842229156589546, −5.53155689352443585301060309910, −4.80242626353199395549421361659, −3.64515952200172014326527315180, −3.40685581739085458744058914210, −1.90259305109790850687900049598, −0.932802791679239796291896672128, 0.19658291332506261951459708821, 2.03333071650062339870501312043, 2.72846244587180176851128880031, 3.44553128546543842739066733448, 4.28084606003906274189634196258, 5.26806054955368609284259615181, 6.19098698932304683265207026637, 6.76068904127562868143619413242, 7.27271515840520538974400268484, 7.85035871410912181983390609077

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6285729531\)
\(L(\frac12)\)	\(\approx\)	\(0.6285729531\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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