LMFDB - L-function 2-2205-21.20-c1-0-21

Dirichlet series

L(s) = 1

− 2.65i·2^-s − 5.04·4^-s + 5^-s + 8.08i·8^-s − 2.65i·10^-s − 4.28i·11^-s + 3.72i·13^-s + 11.3·16^-s + 5.64·17^-s + 6.24i·19^-s − 5.04·20^-s − 11.3·22^-s − 2.66i·23^-s + 25^-s + 9.87·26^-s + ⋯

L(s) = 1

− 1.87i·2^-s − 2.52·4^-s + 0.447·5^-s + 2.85i·8^-s − 0.839i·10^-s − 1.29i·11^-s + 1.03i·13^-s + 2.84·16^-s + 1.36·17^-s + 1.43i·19^-s − 1.12·20^-s − 2.42·22^-s − 0.556i·23^-s + 0.200·25^-s + 1.93·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2205$ = $3^{2} \cdot 5 \cdot 7^{2}$
Sign:	$-0.192 + 0.981i$
Analytic conductor:	$17.6070$
Root analytic conductor:	$4.19607$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2205} (881, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2205,\ (\ :1/2),\ -0.192 + 0.981i)$

Particular Values

$L(1)$	$\approx$	$1.584593346$
$L(\frac12)$	$\approx$	$1.584593346$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 $
	5	$ 1 - T $
	7	$ 1 $
good	2	$ 1 + 2.65iT - 2T^{2} $
	11	$ 1 + 4.28iT - 11T^{2} $
	13	$ 1 - 3.72iT - 13T^{2} $
	17	$ 1 - 5.64T + 17T^{2} $
	19	$ 1 - 6.24iT - 19T^{2} $
	23	$ 1 + 2.66iT - 23T^{2} $
	29	$ 1 - 10.7iT - 29T^{2} $
	31	$ 1 - 6.21iT - 31T^{2} $
	37	$ 1 - 1.68T + 37T^{2} $
	41	$ 1 - 4.50T + 41T^{2} $
	43	$ 1 - 4.89T + 43T^{2} $
	47	$ 1 - 7.85T + 47T^{2} $
	53	$ 1 - 9.01iT - 53T^{2} $
	59	$ 1 + 7.79T + 59T^{2} $
	61	$ 1 + 13.7iT - 61T^{2} $
	67	$ 1 - 0.434T + 67T^{2} $
	71	$ 1 + 6.48iT - 71T^{2} $
	73	$ 1 + 7.40iT - 73T^{2} $
	79	$ 1 + 10.6T + 79T^{2} $
	83	$ 1 + 8.90T + 83T^{2} $
	89	$ 1 - 9.92T + 89T^{2} $
	97	$ 1 - 3.04iT - 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

−9.035502216356205069235684420007, −8.564161532633633245874411953380, −7.56730643099943009487598551944, −6.16472899069096424446352319369, −5.45055135052836158101534750767, −4.51213448247896936605296661359, −3.51296191341093088460063361670, −3.02966854918199953011385548240, −1.78227134274926281383646384652, −1.03104087509445307327663039887, 0.72490070734523574375958886497, 2.58978487680974085340904497908, 4.02895633889265116006557211518, 4.74364450901040482261970964270, 5.66001052781197338855746844508, 5.98429177718917774500122272097, 7.15790833365299688519126996934, 7.50836026418636738038669416501, 8.180654500239416977869338118828, 9.161412409442008383882106968679

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.584593346\)
\(L(\frac12)\)	\(\approx\)	\(1.584593346\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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