LMFDB - L-function 2-4000-8.5-c1-0-50

Dirichlet series

L(s) = 1

− 0.207i·3^-s + 2.73·7^-s + 2.95·9^-s + 0.871i·11^-s + 1.98i·13^-s − 2.93·17^-s − 0.905i·19^-s − 0.566i·21^-s + 6.50·23^-s − 1.23i·27^-s + 7.71i·29^-s + 4.14·31^-s + 0.180·33^-s − 0.436i·37^-s + 0.410·39^-s + ⋯

L(s) = 1

− 0.119i·3^-s + 1.03·7^-s + 0.985·9^-s + 0.262i·11^-s + 0.550i·13^-s − 0.711·17^-s − 0.207i·19^-s − 0.123i·21^-s + 1.35·23^-s − 0.237i·27^-s + 1.43i·29^-s + 0.744·31^-s + 0.0314·33^-s − 0.0718i·37^-s + 0.0658·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4000$ = $2^{5} \cdot 5^{3}$
Sign:	$0.914 - 0.404i$
Analytic conductor:	$31.9401$
Root analytic conductor:	$5.65156$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4000} (2001, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4000,\ (\ :1/2),\ 0.914 - 0.404i)$

Particular Values

$L(1)$	$\approx$	$2.433458659$
$L(\frac12)$	$\approx$	$2.433458659$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
good	3	$ 1 + 0.207iT - 3T^{2} $
	7	$ 1 - 2.73T + 7T^{2} $
	11	$ 1 - 0.871iT - 11T^{2} $
	13	$ 1 - 1.98iT - 13T^{2} $
	17	$ 1 + 2.93T + 17T^{2} $
	19	$ 1 + 0.905iT - 19T^{2} $
	23	$ 1 - 6.50T + 23T^{2} $
	29	$ 1 - 7.71iT - 29T^{2} $
	31	$ 1 - 4.14T + 31T^{2} $
	37	$ 1 + 0.436iT - 37T^{2} $
	41	$ 1 - 5.84T + 41T^{2} $
	43	$ 1 + 3.55iT - 43T^{2} $
	47	$ 1 + 4.69T + 47T^{2} $
	53	$ 1 - 9.68iT - 53T^{2} $
	59	$ 1 + 12.8iT - 59T^{2} $
	61	$ 1 - 12.7iT - 61T^{2} $
	67	$ 1 - 10.2iT - 67T^{2} $
	71	$ 1 - 12.9T + 71T^{2} $
	73	$ 1 + 9.76T + 73T^{2} $
	79	$ 1 + 3.02T + 79T^{2} $
	83	$ 1 + 9.88iT - 83T^{2} $
	89	$ 1 + 8.04T + 89T^{2} $
	97	$ 1 - 12.1T + 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

−8.593880663429921169396447574418, −7.63206079089458574386750163491, −7.08370890265030669514385489550, −6.50778521031429310881532530792, −5.34047665093713371822422330200, −4.66213177074947309171826690518, −4.16357486704042919394180793602, −2.92818234273234868835291097545, −1.87763948153467500935592714677, −1.12129887753862768418249273724, 0.831205180424044530980871050196, 1.83899039927304517593549990994, 2.84978079604928203243441224962, 3.95001150558528903784443455017, 4.66139654621753170941361734562, 5.22932332028934208563084803577, 6.26579957935672803529955770655, 6.94166953859460761455679793943, 7.87430953089406589474747637815, 8.178190511834108910949517853070

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.433458659\)
\(L(\frac12)\)	\(\approx\)	\(2.433458659\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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