LMFDB - L-function 2-12e3-8.3-c2-0-31

Dirichlet series

L(s) = 1

+ 3.99i·5^-s − 7.74i·7^-s − 7.68·11^-s + 1.42i·13^-s + 22.3·17^-s − 21.9·19^-s + 18.1i·23^-s + 9.02·25^-s + 15.2i·29^-s + 10.2i·31^-s + 30.9·35^-s − 59.1i·37^-s + 41.3·41^-s − 11.0·43^-s − 63.6i·47^-s + ⋯

L(s) = 1

+ 0.799i·5^-s − 1.10i·7^-s − 0.699·11^-s + 0.109i·13^-s + 1.31·17^-s − 1.15·19^-s + 0.789i·23^-s + 0.360·25^-s + 0.527i·29^-s + 0.329i·31^-s + 0.884·35^-s − 1.59i·37^-s + 1.00·41^-s − 0.255·43^-s − 1.35i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$0.965 + 0.258i$
Analytic conductor:	$47.0845$
Root analytic conductor:	$6.86182$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (1567, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :1),\ 0.965 + 0.258i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.800153362$
$L(\frac12)$	$\approx$	$1.800153362$
$L(2)$		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 $
good	5	$ 1 - 3.99iT - 25T^{2} $
	7	$ 1 + 7.74iT - 49T^{2} $
	11	$ 1 + 7.68T + 121T^{2} $
	13	$ 1 - 1.42iT - 169T^{2} $
	17	$ 1 - 22.3T + 289T^{2} $
	19	$ 1 + 21.9T + 361T^{2} $
	23	$ 1 - 18.1iT - 529T^{2} $
	29	$ 1 - 15.2iT - 841T^{2} $
	31	$ 1 - 10.2iT - 961T^{2} $
	37	$ 1 + 59.1iT - 1.36e3T^{2} $
	41	$ 1 - 41.3T + 1.68e3T^{2} $
	43	$ 1 + 11.0T + 1.84e3T^{2} $
	47	$ 1 + 63.6iT - 2.20e3T^{2} $
	53	$ 1 - 19.2iT - 2.80e3T^{2} $
	59	$ 1 + 16.1T + 3.48e3T^{2} $
	61	$ 1 - 27.7iT - 3.72e3T^{2} $
	67	$ 1 - 60.6T + 4.48e3T^{2} $
	71	$ 1 + 134. iT - 5.04e3T^{2} $
	73	$ 1 - 90.0T + 5.32e3T^{2} $
	79	$ 1 - 12.0iT - 6.24e3T^{2} $
	83	$ 1 - 164.T + 6.88e3T^{2} $
	89	$ 1 - 35.6T + 7.92e3T^{2} $
	97	$ 1 + 147.T + 9.40e3T^{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

−9.187037496131133211882337249929, −8.113883073908815194823484457926, −7.43877625184809259453494970821, −6.89319654799916547596962550944, −5.92387616805120183548946050371, −5.01420169228185993674822809169, −3.91166332920124436154935485466, −3.23344959811161317001909981309, −2.06289048567130608104652228958, −0.65953633221048804331524976978, 0.822609075256474273486912736573, 2.17307358079332710620423906057, 3.03375404025234203456353321004, 4.35084371016725557176455228035, 5.13273725531366158851384628458, 5.81924151200608844452550436327, 6.63120732150041977706652527719, 7.987559981392179807755555443170, 8.254428364148105833563308049414, 9.122056198717350216861208590015

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.800153362\)
\(L(\frac12)\)	\(\approx\)	\(1.800153362\)
\(L(2)\)		not available
\(L(1)\)		not available

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