LMFDB - L-function 2-84e2-21.20-c1-0-31

Dirichlet series

L(s) = 1

− 2.44·5^-s − 1.41i·11^-s + 5.19i·13^-s − 4.89·17^-s + 1.73i·19^-s − 5.65i·23^-s + 0.999·25^-s − 2.82i·29^-s − 1.73i·31^-s − 37^-s + 7.34·41^-s + 43^-s − 12.2·47^-s − 2.82i·53^-s + 3.46i·55^-s + ⋯

L(s) = 1

− 1.09·5^-s − 0.426i·11^-s + 1.44i·13^-s − 1.18·17^-s + 0.397i·19^-s − 1.17i·23^-s + 0.199·25^-s − 0.525i·29^-s − 0.311i·31^-s − 0.164·37^-s + 1.14·41^-s + 0.152·43^-s − 1.78·47^-s − 0.388i·53^-s + 0.467i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7056$ = $2^{4} \cdot 3^{2} \cdot 7^{2}$
Sign:	$0.970 + 0.239i$
Analytic conductor:	$56.3424$
Root analytic conductor:	$7.50616$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7056} (881, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7056,\ (\ :1/2),\ 0.970 + 0.239i)$

Particular Values

$L(1)$	$\approx$	$1.011884089$
$L(\frac12)$	$\approx$	$1.011884089$
$L(\frac{3}{2})$		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 + 2.44T + 5T^{2} $
	11	$ 1 + 1.41iT - 11T^{2} $
	13	$ 1 - 5.19iT - 13T^{2} $
	17	$ 1 + 4.89T + 17T^{2} $
	19	$ 1 - 1.73iT - 19T^{2} $
	23	$ 1 + 5.65iT - 23T^{2} $
	29	$ 1 + 2.82iT - 29T^{2} $
	31	$ 1 + 1.73iT - 31T^{2} $
	37	$ 1 + T + 37T^{2} $
	41	$ 1 - 7.34T + 41T^{2} $
	43	$ 1 - T + 43T^{2} $
	47	$ 1 + 12.2T + 47T^{2} $
	53	$ 1 + 2.82iT - 53T^{2} $
	59	$ 1 - 4.89T + 59T^{2} $
	61	$ 1 - 3.46iT - 61T^{2} $
	67	$ 1 + 11T + 67T^{2} $
	71	$ 1 - 7.07iT - 71T^{2} $
	73	$ 1 - 1.73iT - 73T^{2} $
	79	$ 1 + 5T + 79T^{2} $
	83	$ 1 - 7.34T + 83T^{2} $
	89	$ 1 - 4.89T + 89T^{2} $
	97	$ 1 - 10.3iT - 97T^{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

−7.962009387570369764831278392993, −7.20847376314400960657178026982, −6.56459627256909754509298251541, −6.00101087214743201555625403289, −4.80578024733843153991400655743, −4.27185708452182941284075955598, −3.75489321560554563063931383823, −2.68407722135059078582110205348, −1.82013452896299372363870189061, −0.44814676742372003915600284813, 0.55604654709917980731123316200, 1.82829091952983086724349884586, 2.96010322456764011441976084067, 3.53401996351601933129416758173, 4.41812917201470013019018445818, 5.01025649090455559377146583600, 5.84973251030199592699287336669, 6.68564244844856192287680341016, 7.48892669647246319702842236329, 7.78581941806451653187857993116

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.011884089\)
\(L(\frac12)\)	\(\approx\)	\(1.011884089\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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