LMFDB - L-function 2-2400-8.5-c1-0-24

Dirichlet series

L(s) = 1

− i·3^-s + 4.68·7^-s − 9^-s − 2.29i·11^-s + 4.97i·13^-s + 2.97·17^-s − 2.68i·19^-s − 4.68i·21^-s + 2.68·23^-s + i·27^-s + 2i·29^-s + 6.97·31^-s − 2.29·33^-s − 4.39i·37^-s + 4.97·39^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 1.77·7^-s − 0.333·9^-s − 0.691i·11^-s + 1.38i·13^-s + 0.722·17^-s − 0.616i·19^-s − 1.02i·21^-s + 0.560·23^-s + 0.192i·27^-s + 0.371i·29^-s + 1.25·31^-s − 0.399·33^-s − 0.722i·37^-s + 0.797·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2400$ = $2^{5} \cdot 3 \cdot 5^{2}$
Sign:	$0.880 + 0.474i$
Analytic conductor:	$19.1640$
Root analytic conductor:	$4.37768$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2400} (1201, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2400,\ (\ :1/2),\ 0.880 + 0.474i)$

Particular Values

$L(1)$	$\approx$	$2.336530255$
$L(\frac12)$	$\approx$	$2.336530255$
$L(\frac{3}{2})$		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 + iT $
	5	$ 1 $
good	7	$ 1 - 4.68T + 7T^{2} $
	11	$ 1 + 2.29iT - 11T^{2} $
	13	$ 1 - 4.97iT - 13T^{2} $
	17	$ 1 - 2.97T + 17T^{2} $
	19	$ 1 + 2.68iT - 19T^{2} $
	23	$ 1 - 2.68T + 23T^{2} $
	29	$ 1 - 2iT - 29T^{2} $
	31	$ 1 - 6.97T + 31T^{2} $
	37	$ 1 + 4.39iT - 37T^{2} $
	41	$ 1 + 11.3T + 41T^{2} $
	43	$ 1 - 9.37iT - 43T^{2} $
	47	$ 1 - 7.27T + 47T^{2} $
	53	$ 1 - 2iT - 53T^{2} $
	59	$ 1 + 1.70iT - 59T^{2} $
	61	$ 1 - 4.58iT - 61T^{2} $
	67	$ 1 - 4iT - 67T^{2} $
	71	$ 1 + 0.585T + 71T^{2} $
	73	$ 1 - 6T + 73T^{2} $
	79	$ 1 + 1.02T + 79T^{2} $
	83	$ 1 + 13.3iT - 83T^{2} $
	89	$ 1 - 3.37T + 89T^{2} $
	97	$ 1 - 3.95T + 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

−8.655765655997940802916244900855, −8.219448378764152475228530045940, −7.38660279535395914844400815166, −6.72514741048864707780638001615, −5.74109086383666946174255372895, −4.93010494011419526236306266712, −4.24527281148814458843820607764, −2.95486725235835732514757401708, −1.86508079173314118026685808326, −1.06293926203455231960508260983, 1.07204925848412328350684457097, 2.21771419985665381855141359655, 3.35365829398997773389515184498, 4.33944013175279623640133187751, 5.15616148118448597095795163772, 5.49324020513041678135908382538, 6.77053617513726686836031552801, 7.891732505204195658275810709892, 8.051136585246293190286448773014, 8.909359602796919825780597036311

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.336530255\)
\(L(\frac12)\)	\(\approx\)	\(2.336530255\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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