LMFDB - L-function 2-45e2-5.4-c1-0-64

Dirichlet series

L(s) = 1

− 1.73i·2^-s − 0.999·4^-s − 2i·7^-s − 1.73i·8^-s − 3.46·11^-s − i·13^-s − 3.46·14^-s − 5·16^-s − 5.19i·17^-s − 2·19^-s + 5.99i·22^-s + 3.46i·23^-s − 1.73·26^-s + 1.99i·28^-s − 1.73·29^-s + ⋯

L(s) = 1

− 1.22i·2^-s − 0.499·4^-s − 0.755i·7^-s − 0.612i·8^-s − 1.04·11^-s − 0.277i·13^-s − 0.925·14^-s − 1.25·16^-s − 1.26i·17^-s − 0.458·19^-s + 1.27i·22^-s + 0.722i·23^-s − 0.339·26^-s + 0.377i·28^-s − 0.321·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2025$ = $3^{4} \cdot 5^{2}$
Sign:	$-0.447 - 0.894i$
Analytic conductor:	$16.1697$
Root analytic conductor:	$4.02115$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2025} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2025,\ (\ :1/2),\ -0.447 - 0.894i)$

Particular Values

$L(1)$	$\approx$	$0.8367451332$
$L(\frac12)$	$\approx$	$0.8367451332$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 $
	5	$ 1 $
good	2	$ 1 + 1.73iT - 2T^{2} $
	7	$ 1 + 2iT - 7T^{2} $
	11	$ 1 + 3.46T + 11T^{2} $
	13	$ 1 + iT - 13T^{2} $
	17	$ 1 + 5.19iT - 17T^{2} $
	19	$ 1 + 2T + 19T^{2} $
	23	$ 1 - 3.46iT - 23T^{2} $
	29	$ 1 + 1.73T + 29T^{2} $
	31	$ 1 - 8T + 31T^{2} $
	37	$ 1 - 7iT - 37T^{2} $
	41	$ 1 + 6.92T + 41T^{2} $
	43	$ 1 - 2iT - 43T^{2} $
	47	$ 1 + 6.92iT - 47T^{2} $
	53	$ 1 - 53T^{2} $
	59	$ 1 + 13.8T + 59T^{2} $
	61	$ 1 + 7T + 61T^{2} $
	67	$ 1 - 10iT - 67T^{2} $
	71	$ 1 + 10.3T + 71T^{2} $
	73	$ 1 + 7iT - 73T^{2} $
	79	$ 1 + 2T + 79T^{2} $
	83	$ 1 + 13.8iT - 83T^{2} $
	89	$ 1 - 5.19T + 89T^{2} $
	97	$ 1 + 2iT - 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

−8.832455530923911914586473091155, −7.80848376464542406599212336638, −7.18584362619360875855465935977, −6.28363569961192941036428684525, −5.07325631206401795385858994105, −4.38024644182308467261085656773, −3.26635679337664497482225881798, −2.69384006054574773533303821647, −1.48321088283183700866022534230, −0.28079742699792462762690322188, 1.96365714280001234364633736269, 2.85065386380197381335095136149, 4.29385597405379900920398936898, 5.09816717814197307883664807277, 5.95256969491012050268713111413, 6.36254490457928315677313940755, 7.35251477860364532749020610531, 8.115631938457230520559388568738, 8.553260256015047482922953279984, 9.349199224898288688071826117064

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8367451332\)
\(L(\frac12)\)	\(\approx\)	\(0.8367451332\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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