LMFDB - L-function 2-45e2-5.4-c1-0-18

Dirichlet series

L(s) = 1

− 2.73i·2^-s − 5.46·4^-s − 1.26i·7^-s + 9.46i·8^-s − 2.26·11^-s + 5.46i·13^-s − 3.46·14^-s + 14.9·16^-s + 0.732i·17^-s + 2.46·19^-s + 6.19i·22^-s − 3.46i·23^-s + 14.9·26^-s + 6.92i·28^-s + 7.19·29^-s + ⋯

L(s) = 1

− 1.93i·2^-s − 2.73·4^-s − 0.479i·7^-s + 3.34i·8^-s − 0.683·11^-s + 1.51i·13^-s − 0.925·14^-s + 3.73·16^-s + 0.177i·17^-s + 0.565·19^-s + 1.32i·22^-s − 0.722i·23^-s + 2.92·26^-s + 1.30i·28^-s + 1.33·29^-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

$L(1)$	$\approx$	$1.230774173$
$L(\frac12)$	$\approx$	$1.230774173$
$L(\frac{3}{2})$		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 + 2.73iT - 2T^{2} $
	7	$ 1 + 1.26iT - 7T^{2} $
	11	$ 1 + 2.26T + 11T^{2} $
	13	$ 1 - 5.46iT - 13T^{2} $
	17	$ 1 - 0.732iT - 17T^{2} $
	19	$ 1 - 2.46T + 19T^{2} $
	23	$ 1 + 3.46iT - 23T^{2} $
	29	$ 1 - 7.19T + 29T^{2} $
	31	$ 1 + 3T + 31T^{2} $
	37	$ 1 - 0.732iT - 37T^{2} $
	41	$ 1 - 3.19T + 41T^{2} $
	43	$ 1 - 10.1iT - 43T^{2} $
	47	$ 1 + 5.26iT - 47T^{2} $
	53	$ 1 + 3.26iT - 53T^{2} $
	59	$ 1 - 11.7T + 59T^{2} $
	61	$ 1 - 4T + 61T^{2} $
	67	$ 1 + 3.46iT - 67T^{2} $
	71	$ 1 + 0.267T + 71T^{2} $
	73	$ 1 + 9.66iT - 73T^{2} $
	79	$ 1 + 8.53T + 79T^{2} $
	83	$ 1 - 8.19iT - 83T^{2} $
	89	$ 1 - 5.19T + 89T^{2} $
	97	$ 1 - 7.66iT - 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

−9.108702393501371906596494810488, −8.534849240324132782852055047790, −7.60509577166237279785779411159, −6.45877213061806128903308929115, −5.18921325623164083039062317798, −4.49297902159190795037728037390, −3.79389246189839866561908683237, −2.78937284844986728808227080122, −1.95187557900100299219476778401, −0.818582328824752729143874248231, 0.67932172919178077325894210413, 2.87209326532031997101080514258, 3.94269265691319818706658022766, 5.12309486758904181405329137113, 5.46241090228724010396829275500, 6.15788297013735382563986648169, 7.24187700488180674231379625269, 7.64134246474337645412211849958, 8.449892737290323815332242597776, 8.962195303430732739782459392192

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.230774173\)
\(L(\frac12)\)	\(\approx\)	\(1.230774173\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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