LMFDB - L-function 2-12e3-8.5-c3-0-46

Dirichlet series

L(s) = 1

− 10.3i·5^-s − 21.4·7^-s − 24.7i·11^-s + 21.4i·13^-s + 123.·17^-s − 7i·19^-s + 114.·23^-s + 17·25^-s + 270. i·29^-s − 214.·31^-s + 222. i·35^-s + 235. i·37^-s − 395.·41^-s − 92i·43^-s + 114.·47^-s + ⋯

L(s) = 1

− 0.929i·5^-s − 1.15·7^-s − 0.678i·11^-s + 0.457i·13^-s + 1.76·17^-s − 0.0845i·19^-s + 1.03·23^-s + 0.136·25^-s + 1.73i·29^-s − 1.24·31^-s + 1.07i·35^-s + 1.04i·37^-s − 1.50·41^-s − 0.326i·43^-s + 0.354·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$101.955$
Root analytic conductor:	$10.0972$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (865, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :3/2),\ 0.707 + 0.707i)$

Particular Values

$L(2)$	$\approx$	$1.781894085$
$L(\frac12)$	$\approx$	$1.781894085$
$L(\frac{5}{2})$		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 + 10.3iT - 125T^{2} $
	7	$ 1 + 21.4T + 343T^{2} $
	11	$ 1 + 24.7iT - 1.33e3T^{2} $
	13	$ 1 - 21.4iT - 2.19e3T^{2} $
	17	$ 1 - 123.T + 4.91e3T^{2} $
	19	$ 1 + 7iT - 6.85e3T^{2} $
	23	$ 1 - 114.T + 1.21e4T^{2} $
	29	$ 1 - 270. iT - 2.43e4T^{2} $
	31	$ 1 + 214.T + 2.97e4T^{2} $
	37	$ 1 - 235. iT - 5.06e4T^{2} $
	41	$ 1 + 395.T + 6.89e4T^{2} $
	43	$ 1 + 92iT - 7.95e4T^{2} $
	47	$ 1 - 114.T + 1.03e5T^{2} $
	53	$ 1 + 20.7iT - 1.48e5T^{2} $
	59	$ 1 + 173. iT - 2.05e5T^{2} $
	61	$ 1 - 449. iT - 2.26e5T^{2} $
	67	$ 1 + 353iT - 3.00e5T^{2} $
	71	$ 1 - 789.T + 3.57e5T^{2} $
	73	$ 1 - 425T + 3.89e5T^{2} $
	79	$ 1 - 1.30e3T + 4.93e5T^{2} $
	83	$ 1 + 593. iT - 5.71e5T^{2} $
	89	$ 1 - 74.2T + 7.04e5T^{2} $
	97	$ 1 - 799T + 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

−8.954460078366775328009487293758, −8.234853398335489347020412332046, −7.18582106662189526866065110985, −6.50164572411826178604758809757, −5.45815083840732690927722294793, −4.97245145654246006561787557434, −3.55134342412827757762735936777, −3.14938342042178998345031530558, −1.50620414939785917410177494999, −0.59680120133221078413415178801, 0.70153120918985975560966998515, 2.22211870050629610423795180697, 3.20523770368075345952610976940, 3.70288112691732955625387271526, 5.10821404389914810655081635871, 5.92267827479940008675511908855, 6.74356903619878338279853701437, 7.35517429654050632381732405792, 8.103794860511964078375862233335, 9.356300034881748970971461111833

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.781894085\)
\(L(\frac12)\)	\(\approx\)	\(1.781894085\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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