LMFDB - L-function 2-4600-5.4-c1-0-96

Dirichlet series

L(s) = 1

− 2.95i·3^-s − 3.95i·7^-s − 5.74·9^-s − 0.957·11^-s − 2.74i·13^-s − 5.74i·17^-s + 6.74·19^-s − 11.7·21^-s + i·23^-s + 8.12i·27^-s − 5.21·29^-s − 5.95·31^-s + 2.83i·33^-s − 9.12i·37^-s − 8.12·39^-s + ⋯

L(s) = 1

− 1.70i·3^-s − 1.49i·7^-s − 1.91·9^-s − 0.288·11^-s − 0.761i·13^-s − 1.39i·17^-s + 1.54·19^-s − 2.55·21^-s + 0.208i·23^-s + 1.56i·27^-s − 0.967·29^-s − 1.07·31^-s + 0.493i·33^-s − 1.50i·37^-s − 1.30·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4600$ = $2^{3} \cdot 5^{2} \cdot 23$
Sign:	$-0.447 - 0.894i$
Analytic conductor:	$36.7311$
Root analytic conductor:	$6.06062$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4600} (4049, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4600,\ (\ :1/2),\ -0.447 - 0.894i)$

Particular Values

$L(1)$	$\approx$	$1.375339526$
$L(\frac12)$	$\approx$	$1.375339526$
$L(\frac{3}{2})$		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 $
	23	$ 1 - iT $
good	3	$ 1 + 2.95iT - 3T^{2} $
	7	$ 1 + 3.95iT - 7T^{2} $
	11	$ 1 + 0.957T + 11T^{2} $
	13	$ 1 + 2.74iT - 13T^{2} $
	17	$ 1 + 5.74iT - 17T^{2} $
	19	$ 1 - 6.74T + 19T^{2} $
	29	$ 1 + 5.21T + 29T^{2} $
	31	$ 1 + 5.95T + 31T^{2} $
	37	$ 1 + 9.12iT - 37T^{2} $
	41	$ 1 - 0.252T + 41T^{2} $
	43	$ 1 - 8iT - 43T^{2} $
	47	$ 1 + 5.49iT - 47T^{2} $
	53	$ 1 + 7.12iT - 53T^{2} $
	59	$ 1 - 4.78T + 59T^{2} $
	61	$ 1 - 12.4T + 61T^{2} $
	67	$ 1 + 9.12iT - 67T^{2} $
	71	$ 1 - 1.66T + 71T^{2} $
	73	$ 1 - 12.3iT - 73T^{2} $
	79	$ 1 + 11.8T + 79T^{2} $
	83	$ 1 - 0.704iT - 83T^{2} $
	89	$ 1 - 15.8T + 89T^{2} $
	97	$ 1 - 10.8iT - 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

−7.60749891899960917536263720206, −7.21380932803335607428762624226, −6.76997925782222700329642245181, −5.60330963634968145901546893020, −5.19355378991471582778809749934, −3.82246733063857476341964657819, −3.07569923639357891903380944034, −2.08696144199675852094644105325, −1.00221515237025462256919756739, −0.43215039200325439924888444786, 1.81166787437212012243185777794, 2.83631217822550075622880749205, 3.58146424281946542996058199995, 4.30356250060874856613590651342, 5.27805928863159772762486076728, 5.53390656996627086922856834110, 6.33377654175968675279954334907, 7.50487884450880196741006073596, 8.461226086453458510116896568626, 8.932785431146564220040795085507

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.375339526\)
\(L(\frac12)\)	\(\approx\)	\(1.375339526\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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