LMFDB - L-function 2-48e2-8.3-c2-0-10

Dirichlet series

L(s) = 1

− 8.11i·5^-s − 9.48i·7^-s − 10.5·11^-s + 20.9i·13^-s − 4.92·17^-s + 26.9·19^-s + 43.7i·23^-s − 40.9·25^-s + 14.5i·29^-s + 25.4i·31^-s − 77.0·35^-s + 12.9i·37^-s − 50.1·41^-s + 5.03·43^-s + 40.8i·47^-s + ⋯

L(s) = 1

− 1.62i·5^-s − 1.35i·7^-s − 0.962·11^-s + 1.61i·13^-s − 0.289·17^-s + 1.41·19^-s + 1.90i·23^-s − 1.63·25^-s + 0.500i·29^-s + 0.822i·31^-s − 2.20·35^-s + 0.350i·37^-s − 1.22·41^-s + 0.117·43^-s + 0.869i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2304$ = $2^{8} \cdot 3^{2}$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$62.7794$
Root analytic conductor:	$7.92334$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2304} (127, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2304,\ (\ :1),\ 0.707 - 0.707i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.082490841$
$L(\frac12)$	$\approx$	$1.082490841$
$L(2)$		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 + 8.11iT - 25T^{2} $
	7	$ 1 + 9.48iT - 49T^{2} $
	11	$ 1 + 10.5T + 121T^{2} $
	13	$ 1 - 20.9iT - 169T^{2} $
	17	$ 1 + 4.92T + 289T^{2} $
	19	$ 1 - 26.9T + 361T^{2} $
	23	$ 1 - 43.7iT - 529T^{2} $
	29	$ 1 - 14.5iT - 841T^{2} $
	31	$ 1 - 25.4iT - 961T^{2} $
	37	$ 1 - 12.9iT - 1.36e3T^{2} $
	41	$ 1 + 50.1T + 1.68e3T^{2} $
	43	$ 1 - 5.03T + 1.84e3T^{2} $
	47	$ 1 - 40.8iT - 2.20e3T^{2} $
	53	$ 1 + 27.2iT - 2.80e3T^{2} $
	59	$ 1 - 24.0T + 3.48e3T^{2} $
	61	$ 1 + 28.9iT - 3.72e3T^{2} $
	67	$ 1 + 65.7T + 4.48e3T^{2} $
	71	$ 1 - 90.5iT - 5.04e3T^{2} $
	73	$ 1 - 127.T + 5.32e3T^{2} $
	79	$ 1 - 35.5iT - 6.24e3T^{2} $
	83	$ 1 + 101.T + 6.88e3T^{2} $
	89	$ 1 + 15.6T + 7.92e3T^{2} $
	97	$ 1 - 73.8T + 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

−8.999657519871470369589366755210, −8.131757009228009472922113442997, −7.43260740950310229236840918195, −6.80178104696594223677575428963, −5.46547548472752833948542215616, −4.95646945549695259815571769575, −4.19988336797667450919450758172, −3.36161850196035834893992217441, −1.68539117870889656938406947335, −1.04903794999320851483784304221, 0.28742234718302512122044813532, 2.34769993846576267440093229635, 2.72877056198714328871784590953, 3.46730469590373822378931362385, 4.96731611266981128649391389157, 5.70233238844340791041160156381, 6.28335897589757381073172807337, 7.25012402206449967704147218767, 7.919200837301745114147279604391, 8.586085439300448343428737290517

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.082490841\)
\(L(\frac12)\)	\(\approx\)	\(1.082490841\)
\(L(2)\)		not available
\(L(1)\)		not available

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