LMFDB - L-function 2-825-5.4-c3-0-83

Dirichlet series

L(s) = 1

− 2.82i·2^-s + 3i·3^-s + 0.0470·4^-s + 8.46·6^-s − 7.12i·7^-s − 22.6i·8^-s − 9·9^-s − 11·11^-s + 0.141i·12^-s − 66.8i·13^-s − 20.0·14^-s − 63.6·16^-s − 40.9i·17^-s + 25.3i·18^-s − 4.85·19^-s + ⋯

L(s) = 1

− 0.997i·2^-s + 0.577i·3^-s + 0.00588·4^-s + 0.575·6^-s − 0.384i·7^-s − 1.00i·8^-s − 0.333·9^-s − 0.301·11^-s + 0.00339i·12^-s − 1.42i·13^-s − 0.383·14^-s − 0.994·16^-s − 0.583i·17^-s + 0.332i·18^-s − 0.0585·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$825$ = $3 \cdot 5^{2} \cdot 11$
Sign:	$-0.894 - 0.447i$
Analytic conductor:	$48.6765$
Root analytic conductor:	$6.97686$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{825} (199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 825,\ (\ :3/2),\ -0.894 - 0.447i)$

Particular Values

$L(2)$	$\approx$	$0.9057040850$
$L(\frac12)$	$\approx$	$0.9057040850$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 - 3iT $
	5	$ 1 $
	11	$ 1 + 11T $
good	2	$ 1 + 2.82iT - 8T^{2} $
	7	$ 1 + 7.12iT - 343T^{2} $
	13	$ 1 + 66.8iT - 2.19e3T^{2} $
	17	$ 1 + 40.9iT - 4.91e3T^{2} $
	19	$ 1 + 4.85T + 6.85e3T^{2} $
	23	$ 1 - 128. iT - 1.21e4T^{2} $
	29	$ 1 - 224.T + 2.43e4T^{2} $
	31	$ 1 + 267.T + 2.97e4T^{2} $
	37	$ 1 - 418. iT - 5.06e4T^{2} $
	41	$ 1 + 496.T + 6.89e4T^{2} $
	43	$ 1 - 90.9iT - 7.95e4T^{2} $
	47	$ 1 + 203. iT - 1.03e5T^{2} $
	53	$ 1 + 219. iT - 1.48e5T^{2} $
	59	$ 1 + 585.T + 2.05e5T^{2} $
	61	$ 1 - 156.T + 2.26e5T^{2} $
	67	$ 1 + 638. iT - 3.00e5T^{2} $
	71	$ 1 + 961.T + 3.57e5T^{2} $
	73	$ 1 + 223. iT - 3.89e5T^{2} $
	79	$ 1 + 415.T + 4.93e5T^{2} $
	83	$ 1 + 44.7iT - 5.71e5T^{2} $
	89	$ 1 + 809.T + 7.04e5T^{2} $
	97	$ 1 + 429. iT - 9.12e5T^{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

−9.844685423381918245035656978200, −8.695918223578515994344166034676, −7.70665995091625499381769528373, −6.79377085256022389962062351022, −5.59119159345196476710446798581, −4.67176741746502032444641578450, −3.42229716384985030690189315023, −2.94560931301175477165257764443, −1.49465740158016522337668973524, −0.22036678233296368015150158988, 1.71751595518205255511682060602, 2.61914039559904875412110505461, 4.22714843025704698540806292845, 5.34321469778592266609456092660, 6.18766441563112660325654326056, 6.86113405102495699105089761647, 7.54283542581038382382466449525, 8.592370812542309614808267319451, 8.960014375987630434434787613705, 10.38648626141371545207564788805

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.9057040850\)
\(L(\frac12)\)	\(\approx\)	\(0.9057040850\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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