LMFDB - L-function 2-12e3-24.5-c2-0-15

Dirichlet series

L(s) = 1

− 6·5^-s − 9.94·7^-s − 19.8·11^-s + 9.94i·13^-s + 19.8i·17^-s − 7i·19^-s + 42i·23^-s + 11·25^-s − 24·29^-s − 39.7·31^-s + 59.6·35^-s + 49.7i·37^-s − 39.7i·41^-s − 50i·43^-s + 6i·47^-s + ⋯

L(s) = 1

− 1.20·5^-s − 1.42·7^-s − 1.80·11^-s + 0.765i·13^-s + 1.17i·17^-s − 0.368i·19^-s + 1.82i·23^-s + 0.440·25^-s − 0.827·29^-s − 1.28·31^-s + 1.70·35^-s + 1.34i·37^-s − 0.970i·41^-s − 1.16i·43^-s + 0.127i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$0.707 + 0.707i$
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:	$(2,\ 1728,\ (\ :1),\ 0.707 + 0.707i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.1124796030$
$L(\frac12)$	$\approx$	$0.1124796030$
$L(2)$		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 + 6T + 25T^{2} $
	7	$ 1 + 9.94T + 49T^{2} $
	11	$ 1 + 19.8T + 121T^{2} $
	13	$ 1 - 9.94iT - 169T^{2} $
	17	$ 1 - 19.8iT - 289T^{2} $
	19	$ 1 + 7iT - 361T^{2} $
	23	$ 1 - 42iT - 529T^{2} $
	29	$ 1 + 24T + 841T^{2} $
	31	$ 1 + 39.7T + 961T^{2} $
	37	$ 1 - 49.7iT - 1.36e3T^{2} $
	41	$ 1 + 39.7iT - 1.68e3T^{2} $
	43	$ 1 + 50iT - 1.84e3T^{2} $
	47	$ 1 - 6iT - 2.20e3T^{2} $
	53	$ 1 + 84T + 2.80e3T^{2} $
	59	$ 1 + 19.8T + 3.48e3T^{2} $
	61	$ 1 + 89.5iT - 3.72e3T^{2} $
	67	$ 1 + 53iT - 4.48e3T^{2} $
	71	$ 1 + 60iT - 5.04e3T^{2} $
	73	$ 1 - 119T + 5.32e3T^{2} $
	79	$ 1 + 49.7T + 6.24e3T^{2} $
	83	$ 1 + 6.88e3T^{2} $
	89	$ 1 - 59.6iT - 7.92e3T^{2} $
	97	$ 1 - 13T + 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

−9.115748275088959832516393744181, −8.039584400331662145356192693231, −7.56619474479414846568121641801, −6.76046511890549038284514580421, −5.80172169634818715243801584585, −4.93631296893937077917778973836, −3.65312019550863538503633458515, −3.38734370982682388054088776958, −2.00527760350478434743672981127, −0.079951016934757451988712012269, 0.36864594372157362899833834858, 2.62233141677323860724627646948, 3.14535591473128589976670194378, 4.15246611733144200991494644738, 5.15856833825244790159708941945, 5.98503682229407792174578765532, 7.04976416699571627470251245838, 7.65862370817611677690353488866, 8.262302414115940648020071532784, 9.283345487191047332960228479522

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.1124796030\)
\(L(\frac12)\)	\(\approx\)	\(0.1124796030\)
\(L(2)\)		not available
\(L(1)\)		not available

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