LMFDB - L-function 2-30e2-15.14-c2-0-3

Dirichlet series

L(s) = 1

− 5.48i·7^-s + 9.17i·11^-s + 11.4i·13^-s + 16.9·17^-s − 26.9·19^-s − 4.93·23^-s + 20.5i·29^-s + 20.9·31^-s + 62.4i·37^-s − 40.9i·41^-s + 1.02i·43^-s + 86.2·47^-s + 18.8·49^-s + 96.0·53^-s + 112. i·59^-s + ⋯

L(s) = 1

− 0.783i·7^-s + 0.833i·11^-s + 0.883i·13^-s + 0.998·17^-s − 1.41·19^-s − 0.214·23^-s + 0.707i·29^-s + 0.676·31^-s + 1.68i·37^-s − 0.998i·41^-s + 0.0238i·43^-s + 1.83·47^-s + 0.385·49^-s + 1.81·53^-s + 1.90i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$900$ = $2^{2} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.472 - 0.881i$
Analytic conductor:	$24.5232$
Root analytic conductor:	$4.95209$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{900} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 900,\ (\ :1),\ 0.472 - 0.881i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.575089221$
$L(\frac12)$	$\approx$	$1.575089221$
$L(2)$		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 $
	5	$ 1 $
good	7	$ 1 + 5.48iT - 49T^{2} $
	11	$ 1 - 9.17iT - 121T^{2} $
	13	$ 1 - 11.4iT - 169T^{2} $
	17	$ 1 - 16.9T + 289T^{2} $
	19	$ 1 + 26.9T + 361T^{2} $
	23	$ 1 + 4.93T + 529T^{2} $
	29	$ 1 - 20.5iT - 841T^{2} $
	31	$ 1 - 20.9T + 961T^{2} $
	37	$ 1 - 62.4iT - 1.36e3T^{2} $
	41	$ 1 + 40.9iT - 1.68e3T^{2} $
	43	$ 1 - 1.02iT - 1.84e3T^{2} $
	47	$ 1 - 86.2T + 2.20e3T^{2} $
	53	$ 1 - 96.0T + 2.80e3T^{2} $
	59	$ 1 - 112. iT - 3.48e3T^{2} $
	61	$ 1 + 66.9T + 3.72e3T^{2} $
	67	$ 1 - 76iT - 4.48e3T^{2} $
	71	$ 1 + 24.0iT - 5.04e3T^{2} $
	73	$ 1 + 18.9iT - 5.32e3T^{2} $
	79	$ 1 + 106.T + 6.24e3T^{2} $
	83	$ 1 + 45.1T + 6.88e3T^{2} $
	89	$ 1 - 115. iT - 7.92e3T^{2} $
	97	$ 1 - 87.0iT - 9.40e3T^{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

−10.32738481591861198187221102596, −9.218695056156731554337280916265, −8.413461794159588682719972490698, −7.35280486597116220374176076013, −6.82184925100204446841538012821, −5.74010516442130419751483438037, −4.52475175170665566251511191066, −3.92828802391607416482579294081, −2.47711546948951845089851198538, −1.20591895565844682203200462623, 0.56698035555902834904656178301, 2.22957382444725797769773563285, 3.23554673231470119031588521265, 4.36909592754046658655649101501, 5.71493401879941741142921509911, 5.95852026060672369486507244254, 7.30980717142611946345540427485, 8.256055155576773830523507547077, 8.762891846327224181301874147653, 9.817791181465971683824936629998

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.575089221\)
\(L(\frac12)\)	\(\approx\)	\(1.575089221\)
\(L(2)\)		not available
\(L(1)\)		not available

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