LMFDB - L-function 2-12e3-24.5-c2-0-23

Dirichlet series

L(s) = 1

− 0.519·5^-s − 10.4·7^-s − 14.1·11^-s + 5.39i·13^-s + 24.9i·17^-s − 13.3i·19^-s − 41.1i·23^-s − 24.7·25^-s − 7.85·29^-s + 26.1·31^-s + 5.40·35^-s + 1.53i·37^-s + 21.3i·41^-s + 64.1i·43^-s + 19.8i·47^-s + ⋯

L(s) = 1

− 0.103·5^-s − 1.48·7^-s − 1.28·11^-s + 0.414i·13^-s + 1.46i·17^-s − 0.701i·19^-s − 1.78i·23^-s − 0.989·25^-s − 0.270·29^-s + 0.844·31^-s + 0.154·35^-s + 0.0415i·37^-s + 0.520i·41^-s + 1.49i·43^-s + 0.422i·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:	$\chi_{1728} (161, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :1),\ 0.965 + 0.258i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.9372340778$
$L(\frac12)$	$\approx$	$0.9372340778$
$L(2)$		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 + 0.519T + 25T^{2} $
	7	$ 1 + 10.4T + 49T^{2} $
	11	$ 1 + 14.1T + 121T^{2} $
	13	$ 1 - 5.39iT - 169T^{2} $
	17	$ 1 - 24.9iT - 289T^{2} $
	19	$ 1 + 13.3iT - 361T^{2} $
	23	$ 1 + 41.1iT - 529T^{2} $
	29	$ 1 + 7.85T + 841T^{2} $
	31	$ 1 - 26.1T + 961T^{2} $
	37	$ 1 - 1.53iT - 1.36e3T^{2} $
	41	$ 1 - 21.3iT - 1.68e3T^{2} $
	43	$ 1 - 64.1iT - 1.84e3T^{2} $
	47	$ 1 - 19.8iT - 2.20e3T^{2} $
	53	$ 1 - 68.6T + 2.80e3T^{2} $
	59	$ 1 - 67.8T + 3.48e3T^{2} $
	61	$ 1 + 58.6iT - 3.72e3T^{2} $
	67	$ 1 + 56.1iT - 4.48e3T^{2} $
	71	$ 1 + 107. iT - 5.04e3T^{2} $
	73	$ 1 - 7.73T + 5.32e3T^{2} $
	79	$ 1 + 64.3T + 6.24e3T^{2} $
	83	$ 1 - 125.T + 6.88e3T^{2} $
	89	$ 1 + 59.6iT - 7.92e3T^{2} $
	97	$ 1 + 138.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

−9.117919132186622170306227033685, −8.309927432369926481844176626980, −7.58029394733292967173199518592, −6.39713288476774471703601219194, −6.23347971344264014485982186213, −4.95929946604528529120557840093, −4.03623806457028616004147968172, −3.05024637344282499202037915975, −2.22459037373444500252825174511, −0.44972391133373244047104504665, 0.56335359198558670248481576780, 2.33831039173896838342733559401, 3.17822318717392810512238387937, 3.95999304891116199519326413500, 5.44449832537245254901430122491, 5.66069989610120020683094999599, 6.98395112420508886263354275025, 7.42905210959851218287413038823, 8.355066795240061753644804173930, 9.343051872118776789671193804255

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.9372340778\)
\(L(\frac12)\)	\(\approx\)	\(0.9372340778\)
\(L(2)\)		not available
\(L(1)\)		not available

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