LMFDB - L-function 2-3200-40.29-c1-0-9

Dirichlet series

L(s) = 1

− 2.82·3^-s + 5.00·9^-s + 2.82i·11^-s − 6i·17^-s + 8.48i·19^-s − 5.65·27^-s − 8.00i·33^-s − 6·41^-s + 8.48·43^-s + 7·49^-s + 16.9i·51^-s − 24i·57^-s − 14.1i·59^-s − 8.48·67^-s − 2i·73^-s + ⋯

L(s) = 1

− 1.63·3^-s + 1.66·9^-s + 0.852i·11^-s − 1.45i·17^-s + 1.94i·19^-s − 1.08·27^-s − 1.39i·33^-s − 0.937·41^-s + 1.29·43^-s + 49^-s + 2.37i·51^-s − 3.17i·57^-s − 1.84i·59^-s − 1.03·67^-s − 0.234i·73^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3200$ = $2^{7} \cdot 5^{2}$
Sign:	$-0.447 - 0.894i$
Analytic conductor:	$25.5521$
Root analytic conductor:	$5.05491$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3200} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3200,\ (\ :1/2),\ -0.447 - 0.894i)$

Particular Values

$L(1)$	$\approx$	$0.5548970309$
$L(\frac12)$	$\approx$	$0.5548970309$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
good	3	$ 1 + 2.82T + 3T^{2} $
	7	$ 1 - 7T^{2} $
	11	$ 1 - 2.82iT - 11T^{2} $
	13	$ 1 + 13T^{2} $
	17	$ 1 + 6iT - 17T^{2} $
	19	$ 1 - 8.48iT - 19T^{2} $
	23	$ 1 - 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 + 37T^{2} $
	41	$ 1 + 6T + 41T^{2} $
	43	$ 1 - 8.48T + 43T^{2} $
	47	$ 1 - 47T^{2} $
	53	$ 1 + 53T^{2} $
	59	$ 1 + 14.1iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 + 8.48T + 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 2iT - 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 + 2.82T + 83T^{2} $
	89	$ 1 - 18T + 89T^{2} $
	97	$ 1 - 10iT - 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

−9.093330152497246893976811217682, −7.889463187396321301698784015607, −7.28718095870529808622193294191, −6.54417254541583359713280468730, −5.84254365795268599510982252585, −5.15777430931850633397067813338, −4.53705384143802619088941301196, −3.57668309019101037114579375121, −2.14719491180285972853108765076, −1.01289868854126216052782011381, 0.27587985931878483352762435520, 1.30260998178410283389165241399, 2.71987037431857806562736523410, 3.95709800064914553825214580668, 4.68878493757624640262400859800, 5.54528868499144386230330889876, 6.04164770654386296906493366396, 6.74632939441290087354260362119, 7.43326495932356611527168489400, 8.517061602589050474763396026927

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.5548970309\)
\(L(\frac12)\)	\(\approx\)	\(0.5548970309\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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