LMFDB - L-function 2-3920-28.27-c1-0-32

Dirichlet series

L(s) = 1

− 3.17·3^-s + i·5^-s + 7.10·9^-s + 1.11i·11^-s + 3.17i·13^-s − 3.17i·15^-s + 6.37i·17^-s + 8.30·19^-s − 6.55i·23^-s − 25^-s − 13.0·27^-s + 7.40·29^-s + 1.53·31^-s − 3.55i·33^-s − 0.0147·37^-s + ⋯

L(s) = 1

− 1.83·3^-s + 0.447i·5^-s + 2.36·9^-s + 0.336i·11^-s + 0.881i·13^-s − 0.820i·15^-s + 1.54i·17^-s + 1.90·19^-s − 1.36i·23^-s − 0.200·25^-s − 2.51·27^-s + 1.37·29^-s + 0.276·31^-s − 0.617i·33^-s − 0.00243·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3920$ = $2^{4} \cdot 5 \cdot 7^{2}$
Sign:	$0.409 - 0.912i$
Analytic conductor:	$31.3013$
Root analytic conductor:	$5.59476$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3920} (2351, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3920,\ (\ :1/2),\ 0.409 - 0.912i)$

Particular Values

$L(1)$	$\approx$	$1.039575138$
$L(\frac12)$	$\approx$	$1.039575138$
$L(\frac{3}{2})$		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 $
	5	$ 1 - iT $
	7	$ 1 $
good	3	$ 1 + 3.17T + 3T^{2} $
	11	$ 1 - 1.11iT - 11T^{2} $
	13	$ 1 - 3.17iT - 13T^{2} $
	17	$ 1 - 6.37iT - 17T^{2} $
	19	$ 1 - 8.30T + 19T^{2} $
	23	$ 1 + 6.55iT - 23T^{2} $
	29	$ 1 - 7.40T + 29T^{2} $
	31	$ 1 - 1.53T + 31T^{2} $
	37	$ 1 + 0.0147T + 37T^{2} $
	41	$ 1 - 1.89iT - 41T^{2} $
	43	$ 1 - 6.07iT - 43T^{2} $
	47	$ 1 - 1.68T + 47T^{2} $
	53	$ 1 - 9.65T + 53T^{2} $
	59	$ 1 + 5.50T + 59T^{2} $
	61	$ 1 - 8.26iT - 61T^{2} $
	67	$ 1 + 4.38iT - 67T^{2} $
	71	$ 1 + 14.8iT - 71T^{2} $
	73	$ 1 + 11.2iT - 73T^{2} $
	79	$ 1 + 6.29iT - 79T^{2} $
	83	$ 1 + 7.10T + 83T^{2} $
	89	$ 1 + 11.0iT - 89T^{2} $
	97	$ 1 + 15.3iT - 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

−8.584251747267803807181197970161, −7.55097638039987759183973327440, −6.96015005849013754408581295759, −6.19617917369598110198277844950, −5.91208210319444676148963883313, −4.73322144995122781236219883840, −4.45289793365027575602205443735, −3.25583396691470962854394042305, −1.85211254344958481526019937442, −0.857415588537072648385145607103, 0.63164158408516719211548398004, 1.16827711710905606482989486634, 2.84725665155424365872490554818, 3.91020688705622777664780800774, 5.00118647003951942271085869489, 5.34355280177353730257993879303, 5.80156319327346906344092701638, 6.92479529850654334418751967237, 7.30394490234587361202336455336, 8.182887875026858683413296267702

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.039575138\)
\(L(\frac12)\)	\(\approx\)	\(1.039575138\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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