LMFDB - L-function 2-1692-141.140-c1-0-4

Dirichlet series

L(s) = 1

− 0.213·5^-s + 4.15·7^-s − 4.61·11^-s + 3.89i·13^-s + 3.63i·17^-s + 6.62i·19^-s − 4.75·23^-s − 4.95·25^-s + 3.65·29^-s − 2.43i·31^-s − 0.887·35^-s − 0.809·37^-s − 0.962·41^-s + 3.98i·43^-s + (5.71 + 3.78i)47^-s + ⋯

L(s) = 1

− 0.0954·5^-s + 1.57·7^-s − 1.39·11^-s + 1.07i·13^-s + 0.882i·17^-s + 1.52i·19^-s − 0.991·23^-s − 0.990·25^-s + 0.678·29^-s − 0.436i·31^-s − 0.150·35^-s − 0.133·37^-s − 0.150·41^-s + 0.608i·43^-s + (0.833 + 0.551i)47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1692 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0308 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1692 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0308 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1692$ = $2^{2} \cdot 3^{2} \cdot 47$
Sign:	$-0.0308 - 0.999i$
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:	$(2,\ 1692,\ (\ :1/2),\ -0.0308 - 0.999i)$

Particular Values

$L(1)$	$\approx$	$1.438592641$
$L(\frac12)$	$\approx$	$1.438592641$
$L(\frac{3}{2})$		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 + (-5.71 - 3.78i)T $
good	5	$ 1 + 0.213T + 5T^{2} $
	7	$ 1 - 4.15T + 7T^{2} $
	11	$ 1 + 4.61T + 11T^{2} $
	13	$ 1 - 3.89iT - 13T^{2} $
	17	$ 1 - 3.63iT - 17T^{2} $
	19	$ 1 - 6.62iT - 19T^{2} $
	23	$ 1 + 4.75T + 23T^{2} $
	29	$ 1 - 3.65T + 29T^{2} $
	31	$ 1 + 2.43iT - 31T^{2} $
	37	$ 1 + 0.809T + 37T^{2} $
	41	$ 1 + 0.962T + 41T^{2} $
	43	$ 1 - 3.98iT - 43T^{2} $
	53	$ 1 - 1.10iT - 53T^{2} $
	59	$ 1 - 3.65iT - 59T^{2} $
	61	$ 1 - 5.35T + 61T^{2} $
	67	$ 1 + 2.32iT - 67T^{2} $
	71	$ 1 + 7.00iT - 71T^{2} $
	73	$ 1 - 4.95iT - 73T^{2} $
	79	$ 1 + 2.15T + 79T^{2} $
	83	$ 1 + 0.809iT - 83T^{2} $
	89	$ 1 - 14.5iT - 89T^{2} $
	97	$ 1 - 2.36T + 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

−9.631628588131591129063717584858, −8.405770140164666173863938679722, −8.061859829916145958087267287087, −7.46328839294877532579909641591, −6.16388195099336881831191793284, −5.48670181348679685669941156924, −4.52705016916743654437993530109, −3.86723346476810703044293007431, −2.30624317370840787851571819071, −1.57192805808728625540563072118, 0.53153310615132348783293162840, 2.09599442120510382755376745268, 2.92659089738831709783289392690, 4.31391283063577822525666183470, 5.15455242850551885165460354593, 5.53221069194729513994133493352, 6.95203068631306458084085684029, 7.77661101953326240748064841277, 8.147372984481378254357083689245, 8.987618852682712080885559792780

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.438592641\)
\(L(\frac12)\)	\(\approx\)	\(1.438592641\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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