LMFDB - L-function 2-56e2-8.5-c1-0-4

Dirichlet series

L(s) = 1

− 2i·3^-s + 4i·5^-s − 9^-s − 2i·11^-s + 4i·13^-s + 8·15^-s − 2·17^-s + 6i·19^-s − 11·25^-s − 4i·27^-s + 8i·29^-s − 8·31^-s − 4·33^-s − 8i·37^-s + 8·39^-s + ⋯

L(s) = 1

− 1.15i·3^-s + 1.78i·5^-s − 0.333·9^-s − 0.603i·11^-s + 1.10i·13^-s + 2.06·15^-s − 0.485·17^-s + 1.37i·19^-s − 2.20·25^-s − 0.769i·27^-s + 1.48i·29^-s − 1.43·31^-s − 0.696·33^-s − 1.31i·37^-s + 1.28·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3136$ = $2^{6} \cdot 7^{2}$
Sign:	$-0.707 - 0.707i$
Analytic conductor:	$25.0410$
Root analytic conductor:	$5.00410$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3136} (1569, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3136,\ (\ :1/2),\ -0.707 - 0.707i)$

Particular Values

$L(1)$	$\approx$	$0.7538317069$
$L(\frac12)$	$\approx$	$0.7538317069$
$L(\frac{3}{2})$		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 $
	7	$ 1 $
good	3	$ 1 + 2iT - 3T^{2} $
	5	$ 1 - 4iT - 5T^{2} $
	11	$ 1 + 2iT - 11T^{2} $
	13	$ 1 - 4iT - 13T^{2} $
	17	$ 1 + 2T + 17T^{2} $
	19	$ 1 - 6iT - 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 - 8iT - 29T^{2} $
	31	$ 1 + 8T + 31T^{2} $
	37	$ 1 + 8iT - 37T^{2} $
	41	$ 1 + 10T + 41T^{2} $
	43	$ 1 + 2iT - 43T^{2} $
	47	$ 1 - 8T + 47T^{2} $
	53	$ 1 - 53T^{2} $
	59	$ 1 + 10iT - 59T^{2} $
	61	$ 1 + 4iT - 61T^{2} $
	67	$ 1 + 2iT - 67T^{2} $
	71	$ 1 + 8T + 71T^{2} $
	73	$ 1 - 6T + 73T^{2} $
	79	$ 1 + 8T + 79T^{2} $
	83	$ 1 - 6iT - 83T^{2} $
	89	$ 1 + 10T + 89T^{2} $
	97	$ 1 + 2T + 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

−8.844052824919015478473922825822, −7.973054447944297521680587322391, −7.18424846256551442075010319406, −6.89777273684676876229222352704, −6.23955542235521637674826328128, −5.49087489097460117718802335759, −3.95896742275163387006540266651, −3.35669602939714937477875054702, −2.24171014298562549466593199864, −1.64809380664779852099762856948, 0.22285984564501144949452926465, 1.49116477728180133801822478512, 2.82066639636964323817884690767, 4.05809432186262684661099157940, 4.49660577212233251391361865573, 5.17838310942420548908147759629, 5.69746300385692983490776747026, 6.99139283163495535477572071510, 7.87696500668779724570601459936, 8.690876393445877258299071800535

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7538317069\)
\(L(\frac12)\)	\(\approx\)	\(0.7538317069\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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