LMFDB - L-function 2-475-19.18-c2-0-40

Dirichlet series

L(s) = 1

− 1.40i·2^-s + 2.91i·3^-s + 2.02·4^-s + 4.10·6^-s + 11.7·7^-s − 8.46i·8^-s + 0.493·9^-s − 6.36·11^-s + 5.89i·12^-s − 17.7i·13^-s − 16.4i·14^-s − 3.82·16^-s + 17.5·17^-s − 0.694i·18^-s + (−16.8 − 8.86i)19^-s + ⋯

L(s) = 1

− 0.703i·2^-s + 0.972i·3^-s + 0.505·4^-s + 0.683·6^-s + 1.67·7^-s − 1.05i·8^-s + 0.0548·9^-s − 0.578·11^-s + 0.491i·12^-s − 1.36i·13^-s − 1.17i·14^-s − 0.238·16^-s + 1.03·17^-s − 0.0385i·18^-s + (−0.884 − 0.466i)19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$475$ = $5^{2} \cdot 19$
Sign:	$0.884 + 0.466i$
Analytic conductor:	$12.9428$
Root analytic conductor:	$3.59761$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{475} (151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 475,\ (\ :1),\ 0.884 + 0.466i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.531124113$
$L(\frac12)$	$\approx$	$2.531124113$
$L(2)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	5	$ 1 $
	19	$ 1 + (16.8 + 8.86i)T $
good	2	$ 1 + 1.40iT - 4T^{2} $
	3	$ 1 - 2.91iT - 9T^{2} $
	7	$ 1 - 11.7T + 49T^{2} $
	11	$ 1 + 6.36T + 121T^{2} $
	13	$ 1 + 17.7iT - 169T^{2} $
	17	$ 1 - 17.5T + 289T^{2} $
	23	$ 1 + 1.53T + 529T^{2} $
	29	$ 1 - 39.9iT - 841T^{2} $
	31	$ 1 + 9.24iT - 961T^{2} $
	37	$ 1 + 56.4iT - 1.36e3T^{2} $
	41	$ 1 - 39.5iT - 1.68e3T^{2} $
	43	$ 1 + 18.4T + 1.84e3T^{2} $
	47	$ 1 - 61.5T + 2.20e3T^{2} $
	53	$ 1 - 66.5iT - 2.80e3T^{2} $
	59	$ 1 - 9.42iT - 3.48e3T^{2} $
	61	$ 1 + 36.5T + 3.72e3T^{2} $
	67	$ 1 + 2.55iT - 4.48e3T^{2} $
	71	$ 1 - 53.8iT - 5.04e3T^{2} $
	73	$ 1 - 115.T + 5.32e3T^{2} $
	79	$ 1 + 95.9iT - 6.24e3T^{2} $
	83	$ 1 + 143.T + 6.88e3T^{2} $
	89	$ 1 - 19.4iT - 7.92e3T^{2} $
	97	$ 1 - 163. 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

−10.59820959352926309243186520325, −10.38123000363187899910801027400, −9.144152286550623173507934973388, −8.009199087674056788546993623363, −7.32293683969830873890723355585, −5.66856988449904758352505937649, −4.85719898977099108614135511836, −3.78608534525998527735241906575, −2.60053642185780321708299708485, −1.23073414611091768667503507799, 1.52732556949284498795146647539, 2.24575065261220222288493668385, 4.34385701649595412792385560137, 5.40988481117169994172956810307, 6.43090005530770517829861197569, 7.26229572966342682663620658633, 7.990296698601239941384804252353, 8.460706622664666108258165857703, 10.06842534673059749915215686914, 11.10369210620728583985458403337

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.531124113\)
\(L(\frac12)\)	\(\approx\)	\(2.531124113\)
\(L(2)\)		not available
\(L(1)\)		not available

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