LMFDB - L-function 2-2352-7.6-c2-0-32

Dirichlet series

L(s) = 1

− 1.73i·3^-s + 4.38i·5^-s − 2.99·9^-s + 19.5·11^-s + 6.11i·13^-s + 7.59·15^-s − 8.76i·17^-s + 30.3i·19^-s + 24·23^-s + 5.78·25^-s + 5.19i·27^-s + 13.5·29^-s − 28.0i·31^-s − 33.9i·33^-s − 48.5·37^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.876i·5^-s − 0.333·9^-s + 1.78·11^-s + 0.470i·13^-s + 0.506·15^-s − 0.515i·17^-s + 1.59i·19^-s + 1.04·23^-s + 0.231·25^-s + 0.192i·27^-s + 0.468·29^-s − 0.904i·31^-s − 1.02i·33^-s − 1.31·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2352$ = $2^{4} \cdot 3 \cdot 7^{2}$
Sign:	$0.654 - 0.755i$
Analytic conductor:	$64.0873$
Root analytic conductor:	$8.00545$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2352} (97, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2352,\ (\ :1),\ 0.654 - 0.755i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.238758173$
$L(\frac12)$	$\approx$	$2.238758173$
$L(2)$		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 + 1.73iT $
	7	$ 1 $
good	5	$ 1 - 4.38iT - 25T^{2} $
	11	$ 1 - 19.5T + 121T^{2} $
	13	$ 1 - 6.11iT - 169T^{2} $
	17	$ 1 + 8.76iT - 289T^{2} $
	19	$ 1 - 30.3iT - 361T^{2} $
	23	$ 1 - 24T + 529T^{2} $
	29	$ 1 - 13.5T + 841T^{2} $
	31	$ 1 + 28.0iT - 961T^{2} $
	37	$ 1 + 48.5T + 1.36e3T^{2} $
	41	$ 1 + 7.14iT - 1.68e3T^{2} $
	43	$ 1 + 53.7T + 1.84e3T^{2} $
	47	$ 1 - 39.9iT - 2.20e3T^{2} $
	53	$ 1 + 61.5T + 2.80e3T^{2} $
	59	$ 1 - 77.1iT - 3.48e3T^{2} $
	61	$ 1 - 0.431iT - 3.72e3T^{2} $
	67	$ 1 - 56.9T + 4.48e3T^{2} $
	71	$ 1 - 123.T + 5.04e3T^{2} $
	73	$ 1 + 35.8iT - 5.32e3T^{2} $
	79	$ 1 - 52.1T + 6.24e3T^{2} $
	83	$ 1 + 136. iT - 6.88e3T^{2} $
	89	$ 1 - 15.2iT - 7.92e3T^{2} $
	97	$ 1 - 34.1iT - 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

−8.954738504059058878922471800387, −8.058236424972238995414268573245, −7.19078742882202897060386778203, −6.60307637554926958635121863762, −6.14734372291068335904887733186, −4.97171627564362993168882002685, −3.84978226035777102609205426336, −3.18907588121943278356054450798, −1.98840979755794598100597534865, −1.09235107499025842299297622627, 0.62987352809032726427524837208, 1.60693560935444820782152954596, 3.08717856019047749180231636666, 3.85895733242819000832047960484, 4.86957629712444937724315877341, 5.18886114873155789990219849164, 6.56111094057089963473172868096, 6.88214688215252893418553393709, 8.314956737330268374685140935889, 8.764591997471838479261155354226

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.238758173\)
\(L(\frac12)\)	\(\approx\)	\(2.238758173\)
\(L(2)\)		not available
\(L(1)\)		not available

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