LMFDB - L-function 2-12e3-8.3-c2-0-63

Dirichlet series

L(s) = 1

− 6.63i·5^-s − 13.4i·7^-s − 18.4·11^-s − 18.9·25^-s − 29.3i·29^-s − 61.4i·31^-s − 89.4·35^-s − 132.·49^-s + 105. i·53^-s + 122. i·55^-s + 117.·59^-s + 143.·73^-s + 248. i·77^-s − 58i·79^-s − 165.·83^-s + ⋯

L(s) = 1

− 1.32i·5^-s − 1.92i·7^-s − 1.67·11^-s − 0.758·25^-s − 1.01i·29^-s − 1.98i·31^-s − 2.55·35^-s − 2.71·49^-s + 1.99i·53^-s + 2.22i·55^-s + 1.99·59^-s + 1.96·73^-s + 3.23i·77^-s − 0.734i·79^-s − 1.98·83^-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:	$\chi_{1728} (1567, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :1),\ -0.707 - 0.707i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.9200338645$
$L(\frac12)$	$\approx$	$0.9200338645$
$L(2)$		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 + 6.63iT - 25T^{2} $
	7	$ 1 + 13.4iT - 49T^{2} $
	11	$ 1 + 18.4T + 121T^{2} $
	13	$ 1 - 169T^{2} $
	17	$ 1 + 289T^{2} $
	19	$ 1 + 361T^{2} $
	23	$ 1 - 529T^{2} $
	29	$ 1 + 29.3iT - 841T^{2} $
	31	$ 1 + 61.4iT - 961T^{2} $
	37	$ 1 - 1.36e3T^{2} $
	41	$ 1 + 1.68e3T^{2} $
	43	$ 1 + 1.84e3T^{2} $
	47	$ 1 - 2.20e3T^{2} $
	53	$ 1 - 105. iT - 2.80e3T^{2} $
	59	$ 1 - 117.T + 3.48e3T^{2} $
	61	$ 1 - 3.72e3T^{2} $
	67	$ 1 + 4.48e3T^{2} $
	71	$ 1 - 5.04e3T^{2} $
	73	$ 1 - 143.T + 5.32e3T^{2} $
	79	$ 1 + 58iT - 6.24e3T^{2} $
	83	$ 1 + 165.T + 6.88e3T^{2} $
	89	$ 1 + 7.92e3T^{2} $
	97	$ 1 - 128.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

−8.475969012776861279342415559234, −7.77174818194062451736951000899, −7.39793098302283196904375365547, −6.13880432051641580775514142678, −5.18694342241594614285157440266, −4.47171985579669143614045262240, −3.81763647901023246962025327102, −2.39550814104313377710871444034, −0.987190930114747468350951789690, −0.27469055374492088567566089264, 2.04010529517000824752362310769, 2.75684779485349219653072775597, 3.33908590069716866549716745599, 5.12876994734724786786549407145, 5.42796900465872366107340795070, 6.49226646980777459113875287998, 7.12835143813770675967687164779, 8.218601955460601789227695407983, 8.661018832761980983147355736189, 9.731197487201931588337897663737

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.9200338645\)
\(L(\frac12)\)	\(\approx\)	\(0.9200338645\)
\(L(2)\)		not available
\(L(1)\)		not available

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