LMFDB - L-function 2-4032-21.20-c1-0-44

Dirichlet series

L(s) = 1

+ 2.64·7^-s + 0.913i·11^-s − 9.39i·23^-s − 5·25^-s − 6.06i·29^-s + 10.5·37^-s − 5.29·43^-s + 7.00·49^-s − 14.5i·53^-s + 4·67^-s + 7.57i·71^-s + 2.41i·77^-s + 8·79^-s + 17.8i·107^-s + 10.5·109^-s + ⋯

L(s) = 1

+ 0.999·7^-s + 0.275i·11^-s − 1.95i·23^-s − 25^-s − 1.12i·29^-s + 1.73·37^-s − 0.806·43^-s + 49^-s − 1.99i·53^-s + 0.488·67^-s + 0.898i·71^-s + 0.275i·77^-s + 0.900·79^-s + 1.72i·107^-s + 1.01·109^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$0.577 + 0.816i$
Analytic conductor:	$32.1956$
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:	$(2,\ 4032,\ (\ :1/2),\ 0.577 + 0.816i)$

Particular Values

$L(1)$	$\approx$	$1.951890607$
$L(\frac12)$	$\approx$	$1.951890607$
$L(\frac{3}{2})$		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 - 2.64T $
good	5	$ 1 + 5T^{2} $
	11	$ 1 - 0.913iT - 11T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 + 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 + 9.39iT - 23T^{2} $
	29	$ 1 + 6.06iT - 29T^{2} $
	31	$ 1 - 31T^{2} $
	37	$ 1 - 10.5T + 37T^{2} $
	41	$ 1 + 41T^{2} $
	43	$ 1 + 5.29T + 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 + 14.5iT - 53T^{2} $
	59	$ 1 + 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 - 4T + 67T^{2} $
	71	$ 1 - 7.57iT - 71T^{2} $
	73	$ 1 - 73T^{2} $
	79	$ 1 - 8T + 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 + 89T^{2} $
	97	$ 1 - 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

−8.132135127968674252779429243075, −7.87192599021723413818037949630, −6.82159960547203246477290115303, −6.16943208894232226387707562561, −5.26550488290196437403607661252, −4.52011303949045395587626617689, −3.92226420067679130555353128367, −2.60324930068949114297693333655, −1.91894217027452077732245718210, −0.61058385485364190290755026385, 1.14203709059379675895118815030, 1.99565789840016076365660219284, 3.14254935113825269502291734549, 3.99092902762296418545644920640, 4.84983256668793828436023330567, 5.56523131212742323553356733252, 6.22139941370244896502206972460, 7.37973915590716198810911106239, 7.70520333253861322961621094786, 8.500835104156239779323498819047

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.951890607\)
\(L(\frac12)\)	\(\approx\)	\(1.951890607\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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