LMFDB - L-function 2-48e2-24.5-c2-0-29

Dirichlet series

L(s) = 1

− 5.21·5^-s + 2·7^-s − 4.78·11^-s + 1.38i·13^-s − 14.6i·17^-s + 26.7i·19^-s − 18.0i·23^-s + 2.23·25^-s + 25.0·29^-s − 39.5·31^-s − 10.4·35^-s + 26i·37^-s + 28.8i·41^-s + 9.52i·43^-s − 80.2i·47^-s + ⋯

L(s) = 1

− 1.04·5^-s + 0.285·7^-s − 0.434·11^-s + 0.106i·13^-s − 0.863i·17^-s + 1.40i·19^-s − 0.784i·23^-s + 0.0895·25^-s + 0.862·29^-s − 1.27·31^-s − 0.298·35^-s + 0.702i·37^-s + 0.702i·41^-s + 0.221i·43^-s − 1.70i·47^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$2304$ = $2^{8} \cdot 3^{2}$
Sign:	$0.985 + 0.169i$
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:	$(2,\ 2304,\ (\ :1),\ 0.985 + 0.169i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.271077207$
$L(\frac12)$	$\approx$	$1.271077207$
$L(2)$		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 + 5.21T + 25T^{2} $
	7	$ 1 - 2T + 49T^{2} $
	11	$ 1 + 4.78T + 121T^{2} $
	13	$ 1 - 1.38iT - 169T^{2} $
	17	$ 1 + 14.6iT - 289T^{2} $
	19	$ 1 - 26.7iT - 361T^{2} $
	23	$ 1 + 18.0iT - 529T^{2} $
	29	$ 1 - 25.0T + 841T^{2} $
	31	$ 1 + 39.5T + 961T^{2} $
	37	$ 1 - 26iT - 1.36e3T^{2} $
	41	$ 1 - 28.8iT - 1.68e3T^{2} $
	43	$ 1 - 9.52iT - 1.84e3T^{2} $
	47	$ 1 + 80.2iT - 2.20e3T^{2} $
	53	$ 1 + 9.79T + 2.80e3T^{2} $
	59	$ 1 - 73.5T + 3.48e3T^{2} $
	61	$ 1 + 67.5iT - 3.72e3T^{2} $
	67	$ 1 - 102. iT - 4.48e3T^{2} $
	71	$ 1 - 21.9iT - 5.04e3T^{2} $
	73	$ 1 - 140.T + 5.32e3T^{2} $
	79	$ 1 + 0.476T + 6.24e3T^{2} $
	83	$ 1 + 31.3T + 6.88e3T^{2} $
	89	$ 1 + 13.1iT - 7.92e3T^{2} $
	97	$ 1 + 69.2T + 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

−8.458823542990596666617961137494, −8.173015122411822094248927153233, −7.34377441429643195003035597884, −6.62718767713235964212873186533, −5.54865696968738023027415622666, −4.76534628344152299628552219632, −3.92184203900169108086731892334, −3.11666988021609490804000443246, −1.92697887715776879025120874202, −0.54114668434076213522113447643, 0.60805743151469484490424469146, 2.00357150364543914105591488451, 3.16311471943864693828670931889, 3.97295260160436748733414530319, 4.79511193665341246506616106680, 5.61633912796149682135466520033, 6.64622714919642325331470640128, 7.48709767002200436724134187164, 7.956879178653507665349805894582, 8.776092473666894575184693452934

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.271077207\)
\(L(\frac12)\)	\(\approx\)	\(1.271077207\)
\(L(2)\)		not available
\(L(1)\)		not available

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