LMFDB - L-function 2-12e2-3.2-c6-0-3

Dirichlet series

L(s) = 1

− 63.6i·5^-s + 244·7^-s + 2.39e3i·11^-s − 2.72e3·13^-s − 5.89e3i·17^-s + 5.39e3·19^-s + 2.49e3i·23^-s + 1.15e4·25^-s + 4.20e4i·29^-s − 1.01e4·31^-s − 1.55e4i·35^-s + 6.50e4·37^-s + 6.97e4i·41^-s + 4.94e4·43^-s + 1.64e5i·47^-s + ⋯

L(s) = 1

− 0.509i·5^-s + 0.711·7^-s + 1.79i·11^-s − 1.24·13^-s − 1.19i·17^-s + 0.786·19^-s + 0.205i·23^-s + 0.740·25^-s + 1.72i·29^-s − 0.341·31^-s − 0.362i·35^-s + 1.28·37^-s + 1.01i·41^-s + 0.622·43^-s + 1.58i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$144$ = $2^{4} \cdot 3^{2}$
Sign:	$0.577 - 0.816i$
Analytic conductor:	$33.1277$
Root analytic conductor:	$5.75567$
Motivic weight:	$6$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{144} (17, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 144,\ (\ :3),\ 0.577 - 0.816i)$

Particular Values

$L(\frac{7}{2})$	$\approx$	$1.843582496$
$L(\frac12)$	$\approx$	$1.843582496$
$L(4)$		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 + 63.6iT - 1.56e4T^{2} $
	7	$ 1 - 244T + 1.17e5T^{2} $
	11	$ 1 - 2.39e3iT - 1.77e6T^{2} $
	13	$ 1 + 2.72e3T + 4.82e6T^{2} $
	17	$ 1 + 5.89e3iT - 2.41e7T^{2} $
	19	$ 1 - 5.39e3T + 4.70e7T^{2} $
	23	$ 1 - 2.49e3iT - 1.48e8T^{2} $
	29	$ 1 - 4.20e4iT - 5.94e8T^{2} $
	31	$ 1 + 1.01e4T + 8.87e8T^{2} $
	37	$ 1 - 6.50e4T + 2.56e9T^{2} $
	41	$ 1 - 6.97e4iT - 4.75e9T^{2} $
	43	$ 1 - 4.94e4T + 6.32e9T^{2} $
	47	$ 1 - 1.64e5iT - 1.07e10T^{2} $
	53	$ 1 + 2.48e4iT - 2.21e10T^{2} $
	59	$ 1 - 3.59e5iT - 4.21e10T^{2} $
	61	$ 1 - 1.00e5T + 5.15e10T^{2} $
	67	$ 1 - 4.35e5T + 9.04e10T^{2} $
	71	$ 1 + 6.01e4iT - 1.28e11T^{2} $
	73	$ 1 - 6.19e5T + 1.51e11T^{2} $
	79	$ 1 + 5.14e5T + 2.43e11T^{2} $
	83	$ 1 + 2.43e5iT - 3.26e11T^{2} $
	89	$ 1 - 4.28e5iT - 4.96e11T^{2} $
	97	$ 1 - 4.27e4T + 8.32e11T^{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

−12.23909437093049553625119935263, −11.23940373367742370849916580287, −9.860020745158811528311007197193, −9.220340585347905032672667919505, −7.70569731628631915516692443021, −7.06091044702677641275808680522, −5.10283548030469841643446138010, −4.63008835440417881123834252963, −2.61015361811554698029106023033, −1.23088329867627156925572204342, 0.61897037394923263601546122252, 2.36933422790825144710130521220, 3.71448859292368612564012358881, 5.23664017345010824141485313057, 6.33103942886111571040949466708, 7.69057638228113569459024644576, 8.526075741908473970468927427051, 9.860730350973388316365323358594, 10.94694450333794800388663671687, 11.59699758446124471075238294406

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 $\F_{q}$

\(L(\frac{7}{2})\)	\(\approx\)	\(1.843582496\)
\(L(\frac12)\)	\(\approx\)	\(1.843582496\)
\(L(4)\)		not available
\(L(1)\)		not available

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