LMFDB - L-function 2-4032-24.11-c1-0-34

Dirichlet series

L(s) = 1

+ 3.09·5^-s + i·7^-s − 0.646i·11^-s − 4.37i·13^-s − 0.295i·17^-s + 5.29·19^-s + 3.94·23^-s + 4.58·25^-s − 5.03·29^-s + 9.16i·31^-s + 3.09i·35^-s − 2.55i·37^-s − 10.4i·41^-s + 1.63·43^-s + 5.65·47^-s + ⋯

L(s) = 1

+ 1.38·5^-s + 0.377i·7^-s − 0.194i·11^-s − 1.21i·13^-s − 0.0715i·17^-s + 1.21·19^-s + 0.823·23^-s + 0.916·25^-s − 0.934·29^-s + 1.64i·31^-s + 0.523i·35^-s − 0.419i·37^-s − 1.62i·41^-s + 0.249·43^-s + 0.825·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.346i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$0.938 + 0.346i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (2591, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ 0.938 + 0.346i)$

Particular Values

$L(1)$	$\approx$	$2.672892561$
$L(\frac12)$	$\approx$	$2.672892561$
$L(\frac{3}{2})$		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 - iT $
good	5	$ 1 - 3.09T + 5T^{2} $
	11	$ 1 + 0.646iT - 11T^{2} $
	13	$ 1 + 4.37iT - 13T^{2} $
	17	$ 1 + 0.295iT - 17T^{2} $
	19	$ 1 - 5.29T + 19T^{2} $
	23	$ 1 - 3.94T + 23T^{2} $
	29	$ 1 + 5.03T + 29T^{2} $
	31	$ 1 - 9.16iT - 31T^{2} $
	37	$ 1 + 2.55iT - 37T^{2} $
	41	$ 1 + 10.4iT - 41T^{2} $
	43	$ 1 - 1.63T + 43T^{2} $
	47	$ 1 - 5.65T + 47T^{2} $
	53	$ 1 - 8.64T + 53T^{2} $
	59	$ 1 - 59T^{2} $
	61	$ 1 + 6.92iT - 61T^{2} $
	67	$ 1 - 2.55T + 67T^{2} $
	71	$ 1 + 1.11T + 71T^{2} $
	73	$ 1 + 9.58T + 73T^{2} $
	79	$ 1 + 16.7iT - 79T^{2} $
	83	$ 1 - 8.50iT - 83T^{2} $
	89	$ 1 - 10.4iT - 89T^{2} $
	97	$ 1 + 2.41T + 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

−8.625268176607400406730932685583, −7.55222064647235483342162987196, −6.97973369079446454514295356664, −5.92793574849634871239310232295, −5.50079204180040841871972895416, −5.00605107300911761079948680600, −3.56770362408708980483863939893, −2.84646165518699784038249342948, −1.94838550390134719203087879337, −0.880122482303472240044464635947, 1.11144066092977623967537942955, 1.98020939971037556012887490198, 2.83616340736826752750846085432, 3.97015570602627701201767208045, 4.77042010680623732791264680160, 5.64253195822907093500451820381, 6.17069412830034127847430221163, 7.05344619921147804855960833102, 7.54968449734941311873315414754, 8.680694140592326464874584575136

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.672892561\)
\(L(\frac12)\)	\(\approx\)	\(2.672892561\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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