LMFDB - L-function 2-84e2-12.11-c1-0-41

Dirichlet series

L(s) = 1

− 1.41i·5^-s − 6·13^-s + 7.07i·17^-s + 2.99·25^-s − 4.24i·29^-s − 2·37^-s + 1.41i·41^-s − 12.7i·53^-s + 12·61^-s + 8.48i·65^-s + 6·73^-s + 10.0·85^-s + 18.3i·89^-s + 18·97^-s − 15.5i·101^-s + ⋯

L(s) = 1

− 0.632i·5^-s − 1.66·13^-s + 1.71i·17^-s + 0.599·25^-s − 0.787i·29^-s − 0.328·37^-s + 0.220i·41^-s − 1.74i·53^-s + 1.53·61^-s + 1.05i·65^-s + 0.702·73^-s + 1.08·85^-s + 1.94i·89^-s + 1.82·97^-s − 1.54i·101^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7056$ = $2^{4} \cdot 3^{2} \cdot 7^{2}$
Sign:	$0.577 + 0.816i$
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:	$(2,\ 7056,\ (\ :1/2),\ 0.577 + 0.816i)$

Particular Values

$L(1)$	$\approx$	$1.458597269$
$L(\frac12)$	$\approx$	$1.458597269$
$L(\frac{3}{2})$		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 + 1.41iT - 5T^{2} $
	11	$ 1 + 11T^{2} $
	13	$ 1 + 6T + 13T^{2} $
	17	$ 1 - 7.07iT - 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 + 4.24iT - 29T^{2} $
	31	$ 1 - 31T^{2} $
	37	$ 1 + 2T + 37T^{2} $
	41	$ 1 - 1.41iT - 41T^{2} $
	43	$ 1 - 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 + 12.7iT - 53T^{2} $
	59	$ 1 + 59T^{2} $
	61	$ 1 - 12T + 61T^{2} $
	67	$ 1 - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 - 6T + 73T^{2} $
	79	$ 1 - 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 - 18.3iT - 89T^{2} $
	97	$ 1 - 18T + 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

−8.009253831968333635808125866585, −7.11218798708320075634389802421, −6.49962270629021938054076514075, −5.60900396905505559650093140039, −5.00031148750399214517193006083, −4.31909322736510001751412221574, −3.53229561037629840585753773469, −2.45724058420728031294468177237, −1.71451679437416289699726660263, −0.47402924764212930180657442490, 0.76487608081286292668065186274, 2.21362794244526274173414342729, 2.77550747225285505653088259414, 3.52421364174003075451736740308, 4.75745917187047661738024340184, 4.99956563265915660869631633439, 5.96941352191601822635668683781, 6.95513867203704974731434897456, 7.19145910544666498891113445797, 7.81492954648964393522734584028

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.458597269\)
\(L(\frac12)\)	\(\approx\)	\(1.458597269\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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