LMFDB - L-function 2-1692-141.140-c1-0-3

Dirichlet series

L(s) = 1

− 2.84·5^-s − 3.53·7^-s − 4.06·11^-s − 4.24i·13^-s + 5.67i·17^-s + 0.496i·19^-s + 3.35·23^-s + 3.10·25^-s + 3.85·29^-s − 0.715i·31^-s + 10.0·35^-s + 10.1·37^-s − 0.201·41^-s − 9.49i·43^-s + (−3.15 + 6.08i)47^-s + ⋯

L(s) = 1

− 1.27·5^-s − 1.33·7^-s − 1.22·11^-s − 1.17i·13^-s + 1.37i·17^-s + 0.113i·19^-s + 0.700·23^-s + 0.621·25^-s + 0.716·29^-s − 0.128i·31^-s + 1.70·35^-s + 1.66·37^-s − 0.0315·41^-s − 1.44i·43^-s + (−0.460 + 0.887i)47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1692 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1692 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1692$ = $2^{2} \cdot 3^{2} \cdot 47$
Sign:	$0.990 - 0.136i$
Analytic conductor:	$13.5106$
Root analytic conductor:	$3.67568$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1692} (845, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1692,\ (\ :1/2),\ 0.990 - 0.136i)$

Particular Values

$L(1)$	$\approx$	$0.7195226528$
$L(\frac12)$	$\approx$	$0.7195226528$
$L(\frac{3}{2})$		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 $
	47	$ 1 + (3.15 - 6.08i)T $
good	5	$ 1 + 2.84T + 5T^{2} $
	7	$ 1 + 3.53T + 7T^{2} $
	11	$ 1 + 4.06T + 11T^{2} $
	13	$ 1 + 4.24iT - 13T^{2} $
	17	$ 1 - 5.67iT - 17T^{2} $
	19	$ 1 - 0.496iT - 19T^{2} $
	23	$ 1 - 3.35T + 23T^{2} $
	29	$ 1 - 3.85T + 29T^{2} $
	31	$ 1 + 0.715iT - 31T^{2} $
	37	$ 1 - 10.1T + 37T^{2} $
	41	$ 1 + 0.201T + 41T^{2} $
	43	$ 1 + 9.49iT - 43T^{2} $
	53	$ 1 - 3.66iT - 53T^{2} $
	59	$ 1 + 9.26iT - 59T^{2} $
	61	$ 1 - 8.60T + 61T^{2} $
	67	$ 1 - 13.8iT - 67T^{2} $
	71	$ 1 - 4.39iT - 71T^{2} $
	73	$ 1 + 7.98iT - 73T^{2} $
	79	$ 1 - 5.53T + 79T^{2} $
	83	$ 1 + 2.85iT - 83T^{2} $
	89	$ 1 - 7.77iT - 89T^{2} $
	97	$ 1 + 4.96T + 97T^{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

−9.397011960172762334379260072651, −8.248670668574440131243122033828, −7.974410176777568417323203761512, −7.05878865382162661581496368103, −6.15377806308008670619016734035, −5.32852127734006151118489228873, −4.17814283112447701339191146672, −3.37715255467352710987925375862, −2.66409202908218964719721841817, −0.59140789104505726730238876397, 0.52467597757697256279630802032, 2.60520127249378066443753599169, 3.27088279798742499622900846440, 4.32000790320876233315058785862, 5.05594669482318339534588281730, 6.30529857592330041730086808562, 7.02089902978941212302469609383, 7.64601322660076155909881318913, 8.497958105283252646991820372753, 9.420702068098427412839583861255

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7195226528\)
\(L(\frac12)\)	\(\approx\)	\(0.7195226528\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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