LMFDB - L-function 2-42e2-3.2-c2-0-14

Dirichlet series

L(s) = 1

− 4.24i·5^-s + 21.2i·11^-s − 23·13^-s − 16.9i·17^-s + 19^-s + 16.9i·23^-s + 7.00·25^-s − 33.9i·29^-s + 49·31^-s + 17·37^-s − 21.2i·41^-s + 47·43^-s + 38.1i·47^-s − 84.8i·53^-s + 90·55^-s + ⋯

L(s) = 1

− 0.848i·5^-s + 1.92i·11^-s − 1.76·13^-s − 0.998i·17^-s + 0.0526·19^-s + 0.737i·23^-s + 0.280·25^-s − 1.17i·29^-s + 1.58·31^-s + 0.459·37^-s − 0.517i·41^-s + 1.09·43^-s + 0.812i·47^-s − 1.60i·53^-s + 1.63·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1764$ = $2^{2} \cdot 3^{2} \cdot 7^{2}$
Sign:	$0.577 + 0.816i$
Analytic conductor:	$48.0655$
Root analytic conductor:	$6.93293$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1764} (197, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1764,\ (\ :1),\ 0.577 + 0.816i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.577093443$
$L(\frac12)$	$\approx$	$1.577093443$
$L(2)$		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 $
	7	$ 1 $
good	5	$ 1 + 4.24iT - 25T^{2} $
	11	$ 1 - 21.2iT - 121T^{2} $
	13	$ 1 + 23T + 169T^{2} $
	17	$ 1 + 16.9iT - 289T^{2} $
	19	$ 1 - T + 361T^{2} $
	23	$ 1 - 16.9iT - 529T^{2} $
	29	$ 1 + 33.9iT - 841T^{2} $
	31	$ 1 - 49T + 961T^{2} $
	37	$ 1 - 17T + 1.36e3T^{2} $
	41	$ 1 + 21.2iT - 1.68e3T^{2} $
	43	$ 1 - 47T + 1.84e3T^{2} $
	47	$ 1 - 38.1iT - 2.20e3T^{2} $
	53	$ 1 + 84.8iT - 2.80e3T^{2} $
	59	$ 1 + 50.9iT - 3.48e3T^{2} $
	61	$ 1 - 40T + 3.72e3T^{2} $
	67	$ 1 - 23T + 4.48e3T^{2} $
	71	$ 1 - 63.6iT - 5.04e3T^{2} $
	73	$ 1 + 17T + 5.32e3T^{2} $
	79	$ 1 + 79T + 6.24e3T^{2} $
	83	$ 1 + 106. iT - 6.88e3T^{2} $
	89	$ 1 + 135. iT - 7.92e3T^{2} $
	97	$ 1 - 40T + 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

−9.234680549964725380383050013138, −8.089979468417652520522081990235, −7.39105037743402688815214972052, −6.85163895993793674605612552757, −5.52874509450455061412941162874, −4.67908087252188663789356448005, −4.44030665993386275431937822874, −2.75514263683025590001594903457, −1.94857061361561948363931769952, −0.53180230744853940246301521678, 0.872637019290582071025246136265, 2.54703527543721344340552106888, 3.06899211883639223714989313817, 4.19316906988353454958957531765, 5.24429043164878200050976744188, 6.13730835276170523096856146432, 6.76139507243840343052436927540, 7.67767743718618598064271915609, 8.418927520190938847636969774219, 9.162358333194372174501419323104

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.577093443\)
\(L(\frac12)\)	\(\approx\)	\(1.577093443\)
\(L(2)\)		not available
\(L(1)\)		not available

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