LMFDB - L-function 2-4032-24.11-c1-0-45

Dirichlet series

L(s) = 1

− 3.96·5^-s − i·7^-s − 6.41i·11^-s − 5.60i·13^-s − 6.86i·17^-s − 3.46·19^-s + 2.62·23^-s + 10.7·25^-s − 2.44·29^-s + 2i·31^-s + 3.96i·35^-s + 5.60i·37^-s − 6.86i·41^-s + 7.74·43^-s − 49^-s + ⋯

L(s) = 1

− 1.77·5^-s − 0.377i·7^-s − 1.93i·11^-s − 1.55i·13^-s − 1.66i·17^-s − 0.794·19^-s + 0.546·23^-s + 2.14·25^-s − 0.454·29^-s + 0.359i·31^-s + 0.669i·35^-s + 0.921i·37^-s − 1.07i·41^-s + 1.18·43^-s − 0.142·49^-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

$L(1)$	$\approx$	$0.7360097808$
$L(\frac12)$	$\approx$	$0.7360097808$
$L(\frac{3}{2})$		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 + 3.96T + 5T^{2} $
	11	$ 1 + 6.41iT - 11T^{2} $
	13	$ 1 + 5.60iT - 13T^{2} $
	17	$ 1 + 6.86iT - 17T^{2} $
	19	$ 1 + 3.46T + 19T^{2} $
	23	$ 1 - 2.62T + 23T^{2} $
	29	$ 1 + 2.44T + 29T^{2} $
	31	$ 1 - 2iT - 31T^{2} $
	37	$ 1 - 5.60iT - 37T^{2} $
	41	$ 1 + 6.86iT - 41T^{2} $
	43	$ 1 - 7.74T + 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 - 0.578T + 53T^{2} $
	59	$ 1 + 9.79iT - 59T^{2} $
	61	$ 1 + 4.28iT - 61T^{2} $
	67	$ 1 + 5.60T + 67T^{2} $
	71	$ 1 + 11.1T + 71T^{2} $
	73	$ 1 - 7.70T + 73T^{2} $
	79	$ 1 - 11.7iT - 79T^{2} $
	83	$ 1 + 12.8iT - 83T^{2} $
	89	$ 1 - 6.86iT - 89T^{2} $
	97	$ 1 - 4.29T + 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

−8.064920569316097830633043461902, −7.44856765371633319949328648137, −6.76139059059316500663497823262, −5.71321138905301345637199313584, −4.99698503707320685387951818689, −4.09017359965714648529824297905, −3.24272288186909518739447447012, −2.94107082769547959801659746879, −0.77579277753623490958942399402, −0.31449315535691335706024878568, 1.56296419593134142719738644550, 2.45428271183226056574561066463, 3.82137774899681467220708048243, 4.25126534167963289540103918570, 4.71900775061800575632000435915, 6.04065483683060669324953452083, 6.87653293759978088651754563187, 7.39168105803358644811440327709, 8.014347193467429805445608826452, 8.847326011134203878614964640803

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7360097808\)
\(L(\frac12)\)	\(\approx\)	\(0.7360097808\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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