LMFDB - L-function 2-2016-21.20-c1-0-5

Dirichlet series

L(s) = 1

− 3.36·5^-s + (−2.57 + 0.595i)7^-s − 1.64i·11^-s − 3.06i·13^-s − 3.36·17^-s − 5.53i·19^-s + 1.64i·23^-s + 6.29·25^-s + 6.06i·29^-s + 4.33i·31^-s + (8.66 − 2i)35^-s + 7.29·37^-s + 0.979·41^-s + 5.15·43^-s − 11.1·47^-s + ⋯

L(s) = 1

− 1.50·5^-s + (−0.974 + 0.224i)7^-s − 0.496i·11^-s − 0.851i·13^-s − 0.814·17^-s − 1.26i·19^-s + 0.343i·23^-s + 1.25·25^-s + 1.12i·29^-s + 0.779i·31^-s + (1.46 − 0.338i)35^-s + 1.19·37^-s + 0.152·41^-s + 0.786·43^-s − 1.63·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2016$ = $2^{5} \cdot 3^{2} \cdot 7$
Sign:	$0.665 - 0.746i$
Analytic conductor:	$16.0978$
Root analytic conductor:	$4.01221$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2016} (1889, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2016,\ (\ :1/2),\ 0.665 - 0.746i)$

Particular Values

$L(1)$	$\approx$	$0.6697150935$
$L(\frac12)$	$\approx$	$0.6697150935$
$L(\frac{3}{2})$		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 + (2.57 - 0.595i)T $
good	5	$ 1 + 3.36T + 5T^{2} $
	11	$ 1 + 1.64iT - 11T^{2} $
	13	$ 1 + 3.06iT - 13T^{2} $
	17	$ 1 + 3.36T + 17T^{2} $
	19	$ 1 + 5.53iT - 19T^{2} $
	23	$ 1 - 1.64iT - 23T^{2} $
	29	$ 1 - 6.06iT - 29T^{2} $
	31	$ 1 - 4.33iT - 31T^{2} $
	37	$ 1 - 7.29T + 37T^{2} $
	41	$ 1 - 0.979T + 41T^{2} $
	43	$ 1 - 5.15T + 43T^{2} $
	47	$ 1 + 11.1T + 47T^{2} $
	53	$ 1 - 4.24iT - 53T^{2} $
	59	$ 1 + 6.13T + 59T^{2} $
	61	$ 1 - 11.1iT - 61T^{2} $
	67	$ 1 - 8.48T + 67T^{2} $
	71	$ 1 - 5.64iT - 71T^{2} $
	73	$ 1 - 8.11iT - 73T^{2} $
	79	$ 1 + 8.48T + 79T^{2} $
	83	$ 1 - 11.1T + 83T^{2} $
	89	$ 1 - 7.70T + 89T^{2} $
	97	$ 1 - 14.2iT - 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

−9.023760045619278533033393546247, −8.565330653871732222025012628430, −7.63999437182656495117855980540, −7.01184295789142347027082297132, −6.21545479095771544901392445375, −5.15343255756097667129619692104, −4.25813786573505583165638346795, −3.33865745178680784557432971628, −2.75797568915617712953647951815, −0.73486172896466007072849253070, 0.37436380513452813583073892477, 2.13763455261810596141976451661, 3.39708293323859609649989737175, 4.08469683572847376546570161841, 4.66212546963720881816474398204, 6.16049732804927113360171739617, 6.65388675849491654810169389583, 7.67351835667784190316214151396, 8.007578698032090189720099683468, 9.120035609602851841809481689609

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6697150935\)
\(L(\frac12)\)	\(\approx\)	\(0.6697150935\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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