LMFDB - L-function 2-432-12.11-c3-0-11

Dirichlet series

L(s) = 1

− 1.73i·7^-s + 19·13^-s − 29.4i·19^-s + 125·25^-s − 155. i·31^-s + 323·37^-s − 218. i·43^-s + 340·49^-s + 719·61^-s − 1.08e3i·67^-s + 919·73^-s + 219. i·79^-s − 32.9i·91^-s + 523·97^-s − 1.06e3i·103^-s + ⋯

L(s) = 1

− 0.0935i·7^-s + 0.405·13^-s − 0.355i·19^-s + 25^-s − 0.903i·31^-s + 1.43·37^-s − 0.773i·43^-s + 0.991·49^-s + 1.50·61^-s − 1.98i·67^-s + 1.47·73^-s + 0.313i·79^-s − 0.0379i·91^-s + 0.547·97^-s − 1.01i·103^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$432$ = $2^{4} \cdot 3^{3}$
Sign:	$0.866 + 0.5i$
Analytic conductor:	$25.4888$
Root analytic conductor:	$5.04864$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{432} (431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 432,\ (\ :3/2),\ 0.866 + 0.5i)$

Particular Values

$L(2)$	$\approx$	$1.890205073$
$L(\frac12)$	$\approx$	$1.890205073$
$L(\frac{5}{2})$		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 - 125T^{2} $
	7	$ 1 + 1.73iT - 343T^{2} $
	11	$ 1 + 1.33e3T^{2} $
	13	$ 1 - 19T + 2.19e3T^{2} $
	17	$ 1 - 4.91e3T^{2} $
	19	$ 1 + 29.4iT - 6.85e3T^{2} $
	23	$ 1 + 1.21e4T^{2} $
	29	$ 1 - 2.43e4T^{2} $
	31	$ 1 + 155. iT - 2.97e4T^{2} $
	37	$ 1 - 323T + 5.06e4T^{2} $
	41	$ 1 - 6.89e4T^{2} $
	43	$ 1 + 218. iT - 7.95e4T^{2} $
	47	$ 1 + 1.03e5T^{2} $
	53	$ 1 - 1.48e5T^{2} $
	59	$ 1 + 2.05e5T^{2} $
	61	$ 1 - 719T + 2.26e5T^{2} $
	67	$ 1 + 1.08e3iT - 3.00e5T^{2} $
	71	$ 1 + 3.57e5T^{2} $
	73	$ 1 - 919T + 3.89e5T^{2} $
	79	$ 1 - 219. iT - 4.93e5T^{2} $
	83	$ 1 + 5.71e5T^{2} $
	89	$ 1 - 7.04e5T^{2} $
	97	$ 1 - 523T + 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

−10.74764018789133982820915693502, −9.735486889017234240852597847880, −8.867779396507314821856682223617, −7.925985084282022187651443511302, −6.92221803566065232524758835513, −5.95165823242405467129049577517, −4.82049096425990081767150158800, −3.69594475583198303885973765867, −2.37894200608739208586959787653, −0.77821972173069345704641639240, 1.07382989430580318477318560127, 2.62028524574978642979261002363, 3.87049853575799998767040039962, 5.04057753009833443310019609852, 6.10705670820890072819566531428, 7.06633156338691462689860124268, 8.144185094913710438741716766133, 8.954453452551579021295845160875, 9.944035607775396011321778082709, 10.82159686235251345579612532263

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

\(L(2)\)	\(\approx\)	\(1.890205073\)
\(L(\frac12)\)	\(\approx\)	\(1.890205073\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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