LMFDB - L-function 2-48e2-24.5-c2-0-41

Dirichlet series

L(s) = 1

− 7.07·5^-s + 12·7^-s − 5.65·11^-s + 8i·13^-s + 9.89i·17^-s − 16i·19^-s − 39.5i·23^-s + 25.0·25^-s − 29.6·29^-s + 4·31^-s − 84.8·35^-s + 30i·37^-s + 21.2i·41^-s + 8i·43^-s + 16.9i·47^-s + ⋯

L(s) = 1

− 1.41·5^-s + 1.71·7^-s − 0.514·11^-s + 0.615i·13^-s + 0.582i·17^-s − 0.842i·19^-s − 1.72i·23^-s + 1.00·25^-s − 1.02·29^-s + 0.129·31^-s − 2.42·35^-s + 0.810i·37^-s + 0.517i·41^-s + 0.186i·43^-s + 0.361i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2304$ = $2^{8} \cdot 3^{2}$
Sign:	$-0.169 + 0.985i$
Analytic conductor:	$62.7794$
Root analytic conductor:	$7.92334$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2304} (2177, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2304,\ (\ :1),\ -0.169 + 0.985i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.048985014$
$L(\frac12)$	$\approx$	$1.048985014$
$L(2)$		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 $
good	5	$ 1 + 7.07T + 25T^{2} $
	7	$ 1 - 12T + 49T^{2} $
	11	$ 1 + 5.65T + 121T^{2} $
	13	$ 1 - 8iT - 169T^{2} $
	17	$ 1 - 9.89iT - 289T^{2} $
	19	$ 1 + 16iT - 361T^{2} $
	23	$ 1 + 39.5iT - 529T^{2} $
	29	$ 1 + 29.6T + 841T^{2} $
	31	$ 1 - 4T + 961T^{2} $
	37	$ 1 - 30iT - 1.36e3T^{2} $
	41	$ 1 - 21.2iT - 1.68e3T^{2} $
	43	$ 1 - 8iT - 1.84e3T^{2} $
	47	$ 1 - 16.9iT - 2.20e3T^{2} $
	53	$ 1 + 49.4T + 2.80e3T^{2} $
	59	$ 1 - 79.1T + 3.48e3T^{2} $
	61	$ 1 - 14iT - 3.72e3T^{2} $
	67	$ 1 + 88iT - 4.48e3T^{2} $
	71	$ 1 + 28.2iT - 5.04e3T^{2} $
	73	$ 1 - 80T + 5.32e3T^{2} $
	79	$ 1 + 100T + 6.24e3T^{2} $
	83	$ 1 - 130.T + 6.88e3T^{2} $
	89	$ 1 + 148. iT - 7.92e3T^{2} $
	97	$ 1 + 112T + 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

−8.314333360566432553291581252857, −8.023114442984952450674799994033, −7.28084008391663955648048416651, −6.41372869993754606541330088462, −5.12128550886938355512405107694, −4.56767823189433713174181826525, −3.96012697652975078134425815278, −2.70426411494915098623203389753, −1.60310504470730833133508017934, −0.29494889229462058255480816549, 1.04004376914442685464947457849, 2.19143316593271686386545506957, 3.52448194525893440379646586105, 4.10926573972510453010077376445, 5.17990542197228296123252643727, 5.52911379215343922349552272319, 7.11333209525018068889323321587, 7.76391033127584718518320434001, 7.964087468225143919417912350485, 8.786039422381129947249496499541

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.048985014\)
\(L(\frac12)\)	\(\approx\)	\(1.048985014\)
\(L(2)\)		not available
\(L(1)\)		not available

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