LMFDB - L-function 2-12e3-12.11-c3-0-15

Dirichlet series

L(s) = 1

− 3.33i·5^-s + 16.9i·7^-s − 16.8·11^-s − 25.0·13^-s − 116. i·17^-s − 85.4i·19^-s − 158.·23^-s + 113.·25^-s + 269. i·29^-s − 36.0i·31^-s + 56.3·35^-s + 353.·37^-s − 144. i·41^-s + 368. i·43^-s + 397.·47^-s + ⋯

L(s) = 1

− 0.297i·5^-s + 0.912i·7^-s − 0.461·11^-s − 0.535·13^-s − 1.66i·17^-s − 1.03i·19^-s − 1.43·23^-s + 0.911·25^-s + 1.72i·29^-s − 0.209i·31^-s + 0.271·35^-s + 1.57·37^-s − 0.550i·41^-s + 1.30i·43^-s + 1.23·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$-i$
Analytic conductor:	$101.955$
Root analytic conductor:	$10.0972$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (1727, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :3/2),\ -i)$

Particular Values

$L(2)$	$\approx$	$1.187821058$
$L(\frac12)$	$\approx$	$1.187821058$
$L(\frac{5}{2})$		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 + 3.33iT - 125T^{2} $
	7	$ 1 - 16.9iT - 343T^{2} $
	11	$ 1 + 16.8T + 1.33e3T^{2} $
	13	$ 1 + 25.0T + 2.19e3T^{2} $
	17	$ 1 + 116. iT - 4.91e3T^{2} $
	19	$ 1 + 85.4iT - 6.85e3T^{2} $
	23	$ 1 + 158.T + 1.21e4T^{2} $
	29	$ 1 - 269. iT - 2.43e4T^{2} $
	31	$ 1 + 36.0iT - 2.97e4T^{2} $
	37	$ 1 - 353.T + 5.06e4T^{2} $
	41	$ 1 + 144. iT - 6.89e4T^{2} $
	43	$ 1 - 368. iT - 7.95e4T^{2} $
	47	$ 1 - 397.T + 1.03e5T^{2} $
	53	$ 1 + 96.0iT - 1.48e5T^{2} $
	59	$ 1 - 294.T + 2.05e5T^{2} $
	61	$ 1 + 146.T + 2.26e5T^{2} $
	67	$ 1 - 301. iT - 3.00e5T^{2} $
	71	$ 1 + 687.T + 3.57e5T^{2} $
	73	$ 1 + 312.T + 3.89e5T^{2} $
	79	$ 1 - 602. iT - 4.93e5T^{2} $
	83	$ 1 + 1.33e3T + 5.71e5T^{2} $
	89	$ 1 - 856. iT - 7.04e5T^{2} $
	97	$ 1 - 218.T + 9.12e5T^{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

−9.165819451531957973560262140057, −8.497676202666518353580364321832, −7.52926022883070525574462302172, −6.87716343497750540890543329008, −5.76171523050891260149028699538, −5.11010877528394394448751389558, −4.38187268943216033235629913307, −2.89090766244910276977453491829, −2.41551052048171527004548509925, −0.908370860573952625871299080967, 0.29751181847538410347542223918, 1.61893284153425038131109370139, 2.66455625218471255131653717454, 3.92795467700246474101171334932, 4.32061093794092323594191297304, 5.73351306096975581975590242530, 6.23693354785645948278002709188, 7.34174834819864651053767484884, 7.88359902739425958148296716877, 8.589988546002171233737926338131

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 $\F_{q}$
Belyi maps

\(L(2)\)	\(\approx\)	\(1.187821058\)
\(L(\frac12)\)	\(\approx\)	\(1.187821058\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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