LMFDB - L-function 2-12e3-8.3-c2-0-36

Dirichlet series

L(s) = 1

− 3.16i·5^-s + 3.48i·7^-s − 1.13·11^-s + 14.9·25^-s − 29.3i·29^-s + 23.4i·31^-s + 11.0·35^-s + 36.8·49^-s − 56.9i·53^-s + 3.60i·55^-s + 117.·59^-s − 93.7·73^-s − 3.96i·77^-s − 58i·79^-s + 67.0·83^-s + ⋯

L(s) = 1

− 0.633i·5^-s + 0.497i·7^-s − 0.103·11^-s + 0.598·25^-s − 1.01i·29^-s + 0.755i·31^-s + 0.315·35^-s + 0.752·49^-s − 1.07i·53^-s + 0.0655i·55^-s + 1.99·59^-s − 1.28·73^-s − 0.0514i·77^-s − 0.734i·79^-s + 0.807·83^-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

$L(\frac{3}{2})$	$\approx$	$1.838251150$
$L(\frac12)$	$\approx$	$1.838251150$
$L(2)$		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 + 3.16iT - 25T^{2} $
	7	$ 1 - 3.48iT - 49T^{2} $
	11	$ 1 + 1.13T + 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 - 23.4iT - 961T^{2} $
	37	$ 1 - 1.36e3T^{2} $
	41	$ 1 + 1.68e3T^{2} $
	43	$ 1 + 1.84e3T^{2} $
	47	$ 1 - 2.20e3T^{2} $
	53	$ 1 + 56.9iT - 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 + 93.7T + 5.32e3T^{2} $
	79	$ 1 + 58iT - 6.24e3T^{2} $
	83	$ 1 - 67.0T + 6.88e3T^{2} $
	89	$ 1 + 7.92e3T^{2} $
	97	$ 1 - 61.0T + 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

−8.872187373305128112308119364235, −8.456766620751209954326238199508, −7.50708844816533753023217874979, −6.61579215831667185599300182210, −5.69239366974167925944207639758, −4.99353900038615228335703567306, −4.09967948891198661073979496537, −2.96983775772443191495723507365, −1.89968797854150380301015951127, −0.61806009169192532007274278592, 0.922920546563537141627362439046, 2.29988315488131069726556744327, 3.27409818666700371604734190803, 4.16494585156782171368434180698, 5.15584508502648628850107060563, 6.09938031891792078810915343859, 6.96462310184497491668739112505, 7.48385846669784161228249480805, 8.453936475589109492066062377269, 9.232183727983189895237436549811

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.838251150\)
\(L(\frac12)\)	\(\approx\)	\(1.838251150\)
\(L(2)\)		not available
\(L(1)\)		not available

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