LMFDB - L-function 2-4032-3.2-c2-0-89

Dirichlet series

L(s) = 1

− 8.89i·5^-s − 2.64·7^-s + 17.7i·11^-s − 2.58·13^-s − 25.8i·17^-s + 20·19^-s − 17.7i·23^-s − 54.1·25^-s − 11.9i·29^-s + 17.1·31^-s + 23.5i·35^-s − 38·37^-s − 15.7i·41^-s − 43.4·43^-s + 16.9i·47^-s + ⋯

L(s) = 1

− 1.77i·5^-s − 0.377·7^-s + 1.61i·11^-s − 0.198·13^-s − 1.52i·17^-s + 1.05·19^-s − 0.773i·23^-s − 2.16·25^-s − 0.410i·29^-s + 0.553·31^-s + 0.672i·35^-s − 1.02·37^-s − 0.383i·41^-s − 1.01·43^-s + 0.361i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$-0.577 - 0.816i$
Analytic conductor:	$109.864$
Root analytic conductor:	$10.4816$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1),\ -0.577 - 0.816i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.1769100908$
$L(\frac12)$	$\approx$	$0.1769100908$
$L(2)$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	7	$ 1 + 2.64T $
good	5	$ 1 + 8.89iT - 25T^{2} $
	11	$ 1 - 17.7iT - 121T^{2} $
	13	$ 1 + 2.58T + 169T^{2} $
	17	$ 1 + 25.8iT - 289T^{2} $
	19	$ 1 - 20T + 361T^{2} $
	23	$ 1 + 17.7iT - 529T^{2} $
	29	$ 1 + 11.9iT - 841T^{2} $
	31	$ 1 - 17.1T + 961T^{2} $
	37	$ 1 + 38T + 1.36e3T^{2} $
	41	$ 1 + 15.7iT - 1.68e3T^{2} $
	43	$ 1 + 43.4T + 1.84e3T^{2} $
	47	$ 1 - 16.9iT - 2.20e3T^{2} $
	53	$ 1 - 85.5iT - 2.80e3T^{2} $
	59	$ 1 - 1.64iT - 3.48e3T^{2} $
	61	$ 1 + 100.T + 3.72e3T^{2} $
	67	$ 1 - 36.6T + 4.48e3T^{2} $
	71	$ 1 + 17.7iT - 5.04e3T^{2} $
	73	$ 1 - 28.9T + 5.32e3T^{2} $
	79	$ 1 + 118.T + 6.24e3T^{2} $
	83	$ 1 + 120. iT - 6.88e3T^{2} $
	89	$ 1 + 139. iT - 7.92e3T^{2} $
	97	$ 1 - 44.4T + 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

−7.64178783280047639741666861114, −7.32679342930836012622230197333, −6.27958697066649783410346427314, −5.26593888131527828422684475051, −4.79125065688168683419743677866, −4.30110650724531228731790792563, −3.03658948883031071904201245515, −1.97566996288544164018193496725, −1.02914548371527892785166791218, −0.03943090562217877295248466677, 1.48812531604639856567799262090, 2.68202679593457079683530710283, 3.44712053412303322066180759916, 3.67248483361774444961444690742, 5.25800472213124613311182458593, 5.98738834581289042668677446117, 6.52330308019376662586250569744, 7.16506997599767046950049098921, 8.006064882577890224075418148434, 8.583303058230203285117112137831

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.1769100908\)
\(L(\frac12)\)	\(\approx\)	\(0.1769100908\)
\(L(2)\)		not available
\(L(1)\)		not available

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