LMFDB - L-function 2-4032-12.11-c1-0-22

Dirichlet series

L(s) = 1

− 0.665i·5^-s + i·7^-s + 2.07·11^-s − 5.55·13^-s − 2.16i·17^-s + 4.49i·19^-s − 4.28·23^-s + 4.55·25^-s − 1.41i·29^-s − 6.61i·31^-s + 0.665·35^-s + 5.43·37^-s − 5.69i·41^-s − 2.11i·43^-s + 10.5·47^-s + ⋯

L(s) = 1

− 0.297i·5^-s + 0.377i·7^-s + 0.627·11^-s − 1.54·13^-s − 0.524i·17^-s + 1.03i·19^-s − 0.892·23^-s + 0.911·25^-s − 0.262i·29^-s − 1.18i·31^-s + 0.112·35^-s + 0.894·37^-s − 0.889i·41^-s − 0.322i·43^-s + 1.53·47^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$0.816 + 0.577i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (575, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ 0.816 + 0.577i)$

Particular Values

$L(1)$	$\approx$	$1.636533414$
$L(\frac12)$	$\approx$	$1.636533414$
$L(\frac{3}{2})$		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 + 0.665iT - 5T^{2} $
	11	$ 1 - 2.07T + 11T^{2} $
	13	$ 1 + 5.55T + 13T^{2} $
	17	$ 1 + 2.16iT - 17T^{2} $
	19	$ 1 - 4.49iT - 19T^{2} $
	23	$ 1 + 4.28T + 23T^{2} $
	29	$ 1 + 1.41iT - 29T^{2} $
	31	$ 1 + 6.61iT - 31T^{2} $
	37	$ 1 - 5.43T + 37T^{2} $
	41	$ 1 + 5.69iT - 41T^{2} $
	43	$ 1 + 2.11iT - 43T^{2} $
	47	$ 1 - 10.5T + 47T^{2} $
	53	$ 1 - 10.6iT - 53T^{2} $
	59	$ 1 - 13.5T + 59T^{2} $
	61	$ 1 - 0.615T + 61T^{2} $
	67	$ 1 + 14.0iT - 67T^{2} $
	71	$ 1 - 15.5T + 71T^{2} $
	73	$ 1 + 11.9T + 73T^{2} $
	79	$ 1 - 0.824iT - 79T^{2} $
	83	$ 1 + 6.36T + 83T^{2} $
	89	$ 1 - 12.6iT - 89T^{2} $
	97	$ 1 - 0.824T + 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

−8.360060573773636413830536576807, −7.61235166304192396688881058173, −7.02722927791593856247181580061, −6.03433489787495981011384294550, −5.46341489539800674243376720020, −4.54590480520091139118770791005, −3.90328083744405516378406358068, −2.70455296958549551120806528462, −1.98933741999365350588236977846, −0.60285681886471432230095864836, 0.875278314986834915823927579903, 2.17784153476863363437713306591, 2.97189243091398691759292514096, 3.99428340275457118491062983248, 4.71472347883789054495907779576, 5.46924620279390791921142056301, 6.56150049596875855719387815083, 6.96575300745726664277794480518, 7.67811400812115790353488572052, 8.547200039390650689585584996179

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.636533414\)
\(L(\frac12)\)	\(\approx\)	\(1.636533414\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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