LMFDB - L-function 2-84e2-12.11-c1-0-0

Dirichlet series

L(s) = 1

− 2.73i·5^-s + 4.24·11^-s − 6.31·13^-s − 0.732i·17^-s − 6.69i·19^-s − 9.14·23^-s − 2.46·25^-s + 2.44i·29^-s − 1.79i·31^-s − 3.46·37^-s + 4.19i·41^-s + 9.46i·43^-s + 3.46·47^-s + 1.41i·53^-s − 11.5i·55^-s + ⋯

L(s) = 1

− 1.22i·5^-s + 1.27·11^-s − 1.75·13^-s − 0.177i·17^-s − 1.53i·19^-s − 1.90·23^-s − 0.492·25^-s + 0.454i·29^-s − 0.322i·31^-s − 0.569·37^-s + 0.655i·41^-s + 1.44i·43^-s + 0.505·47^-s + 0.194i·53^-s − 1.56i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7056$ = $2^{4} \cdot 3^{2} \cdot 7^{2}$
Sign:	$-0.418 - 0.908i$
Analytic conductor:	$56.3424$
Root analytic conductor:	$7.50616$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7056} (4607, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7056,\ (\ :1/2),\ -0.418 - 0.908i)$

Particular Values

$L(1)$	$\approx$	$0.1185084808$
$L(\frac12)$	$\approx$	$0.1185084808$
$L(\frac{3}{2})$		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 + 2.73iT - 5T^{2} $
	11	$ 1 - 4.24T + 11T^{2} $
	13	$ 1 + 6.31T + 13T^{2} $
	17	$ 1 + 0.732iT - 17T^{2} $
	19	$ 1 + 6.69iT - 19T^{2} $
	23	$ 1 + 9.14T + 23T^{2} $
	29	$ 1 - 2.44iT - 29T^{2} $
	31	$ 1 + 1.79iT - 31T^{2} $
	37	$ 1 + 3.46T + 37T^{2} $
	41	$ 1 - 4.19iT - 41T^{2} $
	43	$ 1 - 9.46iT - 43T^{2} $
	47	$ 1 - 3.46T + 47T^{2} $
	53	$ 1 - 1.41iT - 53T^{2} $
	59	$ 1 + 10.3T + 59T^{2} $
	61	$ 1 + 8.10T + 61T^{2} $
	67	$ 1 - 12.9iT - 67T^{2} $
	71	$ 1 - 4.24T + 71T^{2} $
	73	$ 1 - 8.10T + 73T^{2} $
	79	$ 1 + 8.53iT - 79T^{2} $
	83	$ 1 + 2.53T + 83T^{2} $
	89	$ 1 - 9.66iT - 89T^{2} $
	97	$ 1 - 4.52T + 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

−8.162552724395542801554122752418, −7.52039896978853328407497097126, −6.78176495602107143911677384980, −6.07476173863390433401707986531, −5.14447075437583579525234633693, −4.61849888677119217357460345542, −4.10583719207506506124289562252, −2.92651226854356251685722648879, −2.01778759725552839825193247644, −1.05318654604059890849531074013, 0.02950389597638807702754808242, 1.76100641620780501619224831472, 2.33021492689230113163835176177, 3.44380294472076759868202451055, 3.89989587428836269823285249112, 4.81533206597509666891773549781, 5.86830112460026087887299627204, 6.29426618611803109791478320443, 7.09825900081736793333784517746, 7.54808782091197979962335414204

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.1185084808\)
\(L(\frac12)\)	\(\approx\)	\(0.1185084808\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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