LMFDB - L-function 2-3264-17.16-c1-0-26

Dirichlet series

L(s) = 1

− i·3^-s + 3.60i·5^-s − 4i·7^-s − 9^-s + 3i·11^-s + 13^-s + 3.60·15^-s + (2 + 3.60i)17^-s + 3.60·19^-s − 4·21^-s − 7i·23^-s − 7.99·25^-s + i·27^-s − 7.21i·29^-s + 2i·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 1.61i·5^-s − 1.51i·7^-s − 0.333·9^-s + 0.904i·11^-s + 0.277·13^-s + 0.930·15^-s + (0.485 + 0.874i)17^-s + 0.827·19^-s − 0.872·21^-s − 1.45i·23^-s − 1.59·25^-s + 0.192i·27^-s − 1.33i·29^-s + 0.359i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3264$ = $2^{6} \cdot 3 \cdot 17$
Sign:	$0.874 - 0.485i$
Analytic conductor:	$26.0631$
Root analytic conductor:	$5.10521$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3264} (577, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3264,\ (\ :1/2),\ 0.874 - 0.485i)$

Particular Values

$L(1)$	$\approx$	$1.803630292$
$L(\frac12)$	$\approx$	$1.803630292$
$L(\frac{3}{2})$		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 + iT $
	17	$ 1 + (-2 - 3.60i)T $
good	5	$ 1 - 3.60iT - 5T^{2} $
	7	$ 1 + 4iT - 7T^{2} $
	11	$ 1 - 3iT - 11T^{2} $
	13	$ 1 - T + 13T^{2} $
	19	$ 1 - 3.60T + 19T^{2} $
	23	$ 1 + 7iT - 23T^{2} $
	29	$ 1 + 7.21iT - 29T^{2} $
	31	$ 1 - 2iT - 31T^{2} $
	37	$ 1 - 7.21iT - 37T^{2} $
	41	$ 1 - 10.8iT - 41T^{2} $
	43	$ 1 + 3.60T + 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 - 4T + 53T^{2} $
	59	$ 1 - 14.4T + 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 + 67T^{2} $
	71	$ 1 - 8iT - 71T^{2} $
	73	$ 1 - 73T^{2} $
	79	$ 1 + 4iT - 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 - 12T + 89T^{2} $
	97	$ 1 + 7.21iT - 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

−8.374247205712808368151156336301, −7.75968841831120881833751542036, −7.13888571941631325479851450670, −6.63312572778628798154679315134, −6.08124806272107904385991961192, −4.72864426229455109336794847624, −3.87485800099550600804218691784, −3.12124941639114764390059310227, −2.17428672331473519914695527605, −0.985162184160150530291302550418, 0.68910440224044624790330607034, 1.88864760368510675509188852278, 3.10979092486744393970620329813, 3.86275130477383479128392296149, 5.12691928151590280029262630769, 5.39240957272455196914688803441, 5.81791488746789163213882576182, 7.22738570523916595852554934127, 8.171785788427623670192591178436, 8.777233650745520825554710636498

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.803630292\)
\(L(\frac12)\)	\(\approx\)	\(1.803630292\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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