LMFDB - L-function 2-48e2-3.2-c2-0-44

Dirichlet series

L(s) = 1

− 4.24i·5^-s + 12.6·7^-s − 17.8i·11^-s − 10·13^-s + 24.0i·17^-s + 25.2·19^-s + 17.8i·23^-s + 7.00·25^-s − 15.5i·29^-s + 12.6·31^-s − 53.6i·35^-s + 64·37^-s − 12.7i·41^-s + 50.5·43^-s + 17.8i·47^-s + ⋯

L(s) = 1

− 0.848i·5^-s + 1.80·7^-s − 1.62i·11^-s − 0.769·13^-s + 1.41i·17^-s + 1.33·19^-s + 0.777i·23^-s + 0.280·25^-s − 0.536i·29^-s + 0.408·31^-s − 1.53i·35^-s + 1.72·37^-s − 0.310i·41^-s + 1.17·43^-s + 0.380i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2304$ = $2^{8} \cdot 3^{2}$
Sign:	$0.577 + 0.816i$
Analytic conductor:	$62.7794$
Root analytic conductor:	$7.92334$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2304} (1025, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2304,\ (\ :1),\ 0.577 + 0.816i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.818914224$
$L(\frac12)$	$\approx$	$2.818914224$
$L(2)$		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 + 4.24iT - 25T^{2} $
	7	$ 1 - 12.6T + 49T^{2} $
	11	$ 1 + 17.8iT - 121T^{2} $
	13	$ 1 + 10T + 169T^{2} $
	17	$ 1 - 24.0iT - 289T^{2} $
	19	$ 1 - 25.2T + 361T^{2} $
	23	$ 1 - 17.8iT - 529T^{2} $
	29	$ 1 + 15.5iT - 841T^{2} $
	31	$ 1 - 12.6T + 961T^{2} $
	37	$ 1 - 64T + 1.36e3T^{2} $
	41	$ 1 + 12.7iT - 1.68e3T^{2} $
	43	$ 1 - 50.5T + 1.84e3T^{2} $
	47	$ 1 - 17.8iT - 2.20e3T^{2} $
	53	$ 1 - 18.3iT - 2.80e3T^{2} $
	59	$ 1 - 3.48e3T^{2} $
	61	$ 1 + 3.72e3T^{2} $
	67	$ 1 + 75.8T + 4.48e3T^{2} $
	71	$ 1 + 125. iT - 5.04e3T^{2} $
	73	$ 1 + 96T + 5.32e3T^{2} $
	79	$ 1 + 63.2T + 6.24e3T^{2} $
	83	$ 1 - 125. iT - 6.88e3T^{2} $
	89	$ 1 + 55.1iT - 7.92e3T^{2} $
	97	$ 1 - 64T + 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

−8.685421612069522728546589710731, −7.83028349265146229507771598976, −7.66933678867469604335996659064, −6.06729121048247341815992834340, −5.52109606883229986568312531088, −4.75907504955195506884049680827, −4.01679505943096161871087044185, −2.80196678967592136093360660724, −1.52040837270404785138275962152, −0.842914089828806321312132988969, 1.08215031459177405957993223740, 2.24641112648061256270004100247, 2.86563455356268276913522033211, 4.52702071325911836180804265827, 4.70626373644164298848242106178, 5.65362219212280282001158480086, 7.05409390591970901860904619969, 7.29713376243773631325351325659, 7.918027923107950933888085731866, 8.996568003526001379947017484123

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.818914224\)
\(L(\frac12)\)	\(\approx\)	\(2.818914224\)
\(L(2)\)		not available
\(L(1)\)		not available

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