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

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

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

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

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

$L(\frac{3}{2})$	$\approx$	$3.844677831$
$L(\frac12)$	$\approx$	$3.844677831$
$L(2)$		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

−8.568692954186605935618569849647, −8.322712604793406182130157600189, −7.28686708252666903552511249215, −6.25231970839763618273844285098, −5.72642472179691822187391424798, −4.87321722919007080288302584421, −4.15618141018812305370851210340, −2.68650048211361847914081194775, −1.84442371191743618286855355149, −1.08732325680153495725189712415, 1.22200762364423175530284673438, 1.80870223675713473604432218662, 2.77580557441455400694131724792, 4.20112616256996248208562055703, 5.02434590085314595241418258265, 5.52828290946116994943836104298, 6.52347325686450724512624338938, 7.25169617775908119862479328186, 8.155310006567275362297357389691, 8.971746665892591284179698418210

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(3.844677831\)
\(L(\frac12)\)	\(\approx\)	\(3.844677831\)
\(L(2)\)		not available
\(L(1)\)		not available

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