LMFDB - L-function 2-531-3.2-c2-0-3

Dirichlet series

L(s) = 1

− 0.419i·2^-s + 3.82·4^-s + 8.11i·5^-s − 8.94·7^-s − 3.28i·8^-s + 3.40·10^-s − 4.15i·11^-s − 15.5·13^-s + 3.75i·14^-s + 13.9·16^-s + 29.0i·17^-s − 8.73·19^-s + 31.0i·20^-s − 1.74·22^-s − 0.868i·23^-s + ⋯

L(s) = 1

− 0.209i·2^-s + 0.955·4^-s + 1.62i·5^-s − 1.27·7^-s − 0.410i·8^-s + 0.340·10^-s − 0.377i·11^-s − 1.19·13^-s + 0.268i·14^-s + 0.869·16^-s + 1.70i·17^-s − 0.459·19^-s + 1.55i·20^-s − 0.0793·22^-s − 0.0377i·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$531$ = $3^{2} \cdot 59$
Sign:	$-0.816 - 0.577i$
Analytic conductor:	$14.4687$
Root analytic conductor:	$3.80377$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{531} (296, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 531,\ (\ :1),\ -0.816 - 0.577i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.9516235058$
$L(\frac12)$	$\approx$	$0.9516235058$
$L(2)$		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 $
	59	$ 1 - 7.68iT $
good	2	$ 1 + 0.419iT - 4T^{2} $
	5	$ 1 - 8.11iT - 25T^{2} $
	7	$ 1 + 8.94T + 49T^{2} $
	11	$ 1 + 4.15iT - 121T^{2} $
	13	$ 1 + 15.5T + 169T^{2} $
	17	$ 1 - 29.0iT - 289T^{2} $
	19	$ 1 + 8.73T + 361T^{2} $
	23	$ 1 + 0.868iT - 529T^{2} $
	29	$ 1 + 37.3iT - 841T^{2} $
	31	$ 1 + 35.7T + 961T^{2} $
	37	$ 1 - 19.5T + 1.36e3T^{2} $
	41	$ 1 - 46.2iT - 1.68e3T^{2} $
	43	$ 1 + 77.8T + 1.84e3T^{2} $
	47	$ 1 + 0.219iT - 2.20e3T^{2} $
	53	$ 1 - 63.8iT - 2.80e3T^{2} $
	61	$ 1 - 70.6T + 3.72e3T^{2} $
	67	$ 1 + 56.8T + 4.48e3T^{2} $
	71	$ 1 - 77.4iT - 5.04e3T^{2} $
	73	$ 1 - 118.T + 5.32e3T^{2} $
	79	$ 1 + 53.7T + 6.24e3T^{2} $
	83	$ 1 + 52.8iT - 6.88e3T^{2} $
	89	$ 1 - 104. iT - 7.92e3T^{2} $
	97	$ 1 + 107.T + 9.40e3T^{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

−10.88508696886888893339491676766, −10.19851731475792365816529908811, −9.703921673230812646371919966907, −8.074294868995004657941751512373, −7.14167107439992683472245480601, −6.45722480517016204055284981722, −5.93495802377859579005327165308, −3.82491491438433928911110184341, −3.01373835767463822196620890808, −2.16734656464268039373107530039, 0.32606276880610376064441090794, 2.00835776707711484659545362979, 3.28846984155115582974228734898, 4.83566673988882252381598374586, 5.49269898898119016793811015105, 6.79886838535707436110872815315, 7.35800045620845227699565238389, 8.569524838920666906190801390742, 9.464622426524526238651913069243, 9.998204858765067337240410548720

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.9516235058\)
\(L(\frac12)\)	\(\approx\)	\(0.9516235058\)
\(L(2)\)		not available
\(L(1)\)		not available

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