LMFDB - L-function 2-4032-21.20-c1-0-43

Dirichlet series

L(s) = 1

+ 3.69·5^-s + (2.41 + 1.08i)7^-s + 3.41i·11^-s + 5.22i·13^-s + 6.75·17^-s − 2.16i·19^-s − 6.24i·23^-s + 8.65·25^-s − 2.58i·29^-s − 10.4i·31^-s + (8.92 + 4i)35^-s − 4·37^-s − 0.634·41^-s + 6.48·43^-s − 3.06·47^-s + ⋯

L(s) = 1

+ 1.65·5^-s + (0.912 + 0.409i)7^-s + 1.02i·11^-s + 1.44i·13^-s + 1.63·17^-s − 0.496i·19^-s − 1.30i·23^-s + 1.73·25^-s − 0.480i·29^-s − 1.87i·31^-s + (1.50 + 0.676i)35^-s − 0.657·37^-s − 0.0990·41^-s + 0.988·43^-s − 0.446·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$0.860 - 0.508i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (3905, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ 0.860 - 0.508i)$

Particular Values

$L(1)$	$\approx$	$3.285279236$
$L(\frac12)$	$\approx$	$3.285279236$
$L(\frac{3}{2})$		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 $
	7	$ 1 + (-2.41 - 1.08i)T $
good	5	$ 1 - 3.69T + 5T^{2} $
	11	$ 1 - 3.41iT - 11T^{2} $
	13	$ 1 - 5.22iT - 13T^{2} $
	17	$ 1 - 6.75T + 17T^{2} $
	19	$ 1 + 2.16iT - 19T^{2} $
	23	$ 1 + 6.24iT - 23T^{2} $
	29	$ 1 + 2.58iT - 29T^{2} $
	31	$ 1 + 10.4iT - 31T^{2} $
	37	$ 1 + 4T + 37T^{2} $
	41	$ 1 + 0.634T + 41T^{2} $
	43	$ 1 - 6.48T + 43T^{2} $
	47	$ 1 + 3.06T + 47T^{2} $
	53	$ 1 - 2.58iT - 53T^{2} $
	59	$ 1 + 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 + 1.65T + 67T^{2} $
	71	$ 1 - 7.41iT - 71T^{2} $
	73	$ 1 - 0.896iT - 73T^{2} $
	79	$ 1 + 13.6T + 79T^{2} $
	83	$ 1 + 11.7T + 83T^{2} $
	89	$ 1 - 6.75T + 89T^{2} $
	97	$ 1 + 9.55iT - 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

−8.680606217412387632309781470407, −7.72530775102225198896031791247, −6.99692248817910095872469631833, −6.16245098591976349083084549071, −5.60798766187347630147523687336, −4.79509965423856754615245581685, −4.17806951111218780853496899034, −2.59335683922814285949797779176, −2.08392801932656336778800348132, −1.28906752903050087576803034338, 1.09905629086959116647769805151, 1.61491682266916433842890825191, 2.97804511531472355514182474917, 3.48850718294381624645601311038, 4.97782144276150344654940479305, 5.63076650428021503518918791538, 5.72786405010046386592027608683, 6.92257953278206113996615575227, 7.75340561740539957682608443215, 8.344022784377601522204293017470

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(3.285279236\)
\(L(\frac12)\)	\(\approx\)	\(3.285279236\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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