LMFDB - L-function 2-38e2-19.18-c2-0-53

Dirichlet series

L(s) = 1

− 4.28i·3^-s − 1.88·5^-s + 1.12·7^-s − 9.33·9^-s − 9.34·11^-s − 23.0i·13^-s + 8.08i·15^-s + 15.6·17^-s − 4.79i·21^-s + 10.4·23^-s − 21.4·25^-s + 1.41i·27^-s − 45.5i·29^-s − 43.8i·31^-s + 40.0i·33^-s + ⋯

L(s) = 1

− 1.42i·3^-s − 0.377·5^-s + 0.160·7^-s − 1.03·9^-s − 0.849·11^-s − 1.77i·13^-s + 0.538i·15^-s + 0.918·17^-s − 0.228i·21^-s + 0.454·23^-s − 0.857·25^-s + 0.0524i·27^-s − 1.57i·29^-s − 1.41i·31^-s + 1.21i·33^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1444$ = $2^{2} \cdot 19^{2}$
Sign:	$-0.575 - 0.817i$
Analytic conductor:	$39.3461$
Root analytic conductor:	$6.27265$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1444} (721, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1444,\ (\ :1),\ -0.575 - 0.817i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.7168179844$
$L(\frac12)$	$\approx$	$0.7168179844$
$L(2)$		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 $
	19	$ 1 $
good	3	$ 1 + 4.28iT - 9T^{2} $
	5	$ 1 + 1.88T + 25T^{2} $
	7	$ 1 - 1.12T + 49T^{2} $
	11	$ 1 + 9.34T + 121T^{2} $
	13	$ 1 + 23.0iT - 169T^{2} $
	17	$ 1 - 15.6T + 289T^{2} $
	23	$ 1 - 10.4T + 529T^{2} $
	29	$ 1 + 45.5iT - 841T^{2} $
	31	$ 1 + 43.8iT - 961T^{2} $
	37	$ 1 - 60.6iT - 1.36e3T^{2} $
	41	$ 1 - 48.7iT - 1.68e3T^{2} $
	43	$ 1 - 8.19T + 1.84e3T^{2} $
	47	$ 1 + 71.9T + 2.20e3T^{2} $
	53	$ 1 - 28.3iT - 2.80e3T^{2} $
	59	$ 1 - 58.0iT - 3.48e3T^{2} $
	61	$ 1 + 37.3T + 3.72e3T^{2} $
	67	$ 1 - 55.7iT - 4.48e3T^{2} $
	71	$ 1 + 5.04iT - 5.04e3T^{2} $
	73	$ 1 + 8.53T + 5.32e3T^{2} $
	79	$ 1 + 6.01iT - 6.24e3T^{2} $
	83	$ 1 - 142.T + 6.88e3T^{2} $
	89	$ 1 + 167. iT - 7.92e3T^{2} $
	97	$ 1 + 110. iT - 9.40e3T^{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

−8.272709920505219019022481596603, −7.81941847152568494774342360456, −7.61068771122560509727083498973, −6.29829982304439580946874053791, −5.75281146448970879766799845046, −4.71615464699272893290917329272, −3.27357867635604971920557876130, −2.48767126556477038038614721693, −1.19202644554509273834027710345, −0.21124307215048350849894499933, 1.79034772185594270375397995356, 3.26147484820917320036015445442, 3.89150538753212756419351455933, 4.88802330775880565918205682810, 5.31819541398500493578909601408, 6.63722642322833449399094225950, 7.49794458425634402631679234799, 8.474726331168632992775745449622, 9.236554695702486615801943094416, 9.748353618085162248794625871430

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.7168179844\)
\(L(\frac12)\)	\(\approx\)	\(0.7168179844\)
\(L(2)\)		not available
\(L(1)\)		not available

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