LMFDB - L-function 2-4012-17.16-c1-0-14

Dirichlet series

L(s) = 1

− 2.95i·3^-s + 2.24i·5^-s + 1.49i·7^-s − 5.74·9^-s − 3.01i·11^-s − 1.07·13^-s + 6.63·15^-s + (3.78 − 1.62i)17^-s − 2.42·19^-s + 4.41·21^-s + 4.21i·23^-s − 0.0395·25^-s + 8.11i·27^-s + 7.90i·29^-s − 1.58i·31^-s + ⋯

L(s) = 1

− 1.70i·3^-s + 1.00i·5^-s + 0.564i·7^-s − 1.91·9^-s − 0.908i·11^-s − 0.297·13^-s + 1.71·15^-s + (0.919 − 0.394i)17^-s − 0.556·19^-s + 0.963·21^-s + 0.879i·23^-s − 0.00791·25^-s + 1.56i·27^-s + 1.46i·29^-s − 0.284i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.394i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4012$ = $2^{2} \cdot 17 \cdot 59$
Sign:	$0.919 - 0.394i$
Analytic conductor:	$32.0359$
Root analytic conductor:	$5.66003$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4012} (237, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4012,\ (\ :1/2),\ 0.919 - 0.394i)$

Particular Values

$L(1)$	$\approx$	$1.311097561$
$L(\frac12)$	$\approx$	$1.311097561$
$L(\frac{3}{2})$		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 $
	17	$ 1 + (-3.78 + 1.62i)T $
	59	$ 1 + T $
good	3	$ 1 + 2.95iT - 3T^{2} $
	5	$ 1 - 2.24iT - 5T^{2} $
	7	$ 1 - 1.49iT - 7T^{2} $
	11	$ 1 + 3.01iT - 11T^{2} $
	13	$ 1 + 1.07T + 13T^{2} $
	19	$ 1 + 2.42T + 19T^{2} $
	23	$ 1 - 4.21iT - 23T^{2} $
	29	$ 1 - 7.90iT - 29T^{2} $
	31	$ 1 + 1.58iT - 31T^{2} $
	37	$ 1 - 4.71iT - 37T^{2} $
	41	$ 1 - 3.40iT - 41T^{2} $
	43	$ 1 + 11.0T + 43T^{2} $
	47	$ 1 + 3.19T + 47T^{2} $
	53	$ 1 - 1.46T + 53T^{2} $
	61	$ 1 - 5.45iT - 61T^{2} $
	67	$ 1 + 4.21T + 67T^{2} $
	71	$ 1 + 0.188iT - 71T^{2} $
	73	$ 1 - 8.17iT - 73T^{2} $
	79	$ 1 + 4.28iT - 79T^{2} $
	83	$ 1 - 6.93T + 83T^{2} $
	89	$ 1 - 1.22T + 89T^{2} $
	97	$ 1 - 0.466iT - 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

−8.334803002710818882378078783511, −7.60555089613106383575377913858, −7.03580550444471844048093751903, −6.43116083363924401384636687318, −5.81382763900845409687309304001, −5.06388831440899829067136438644, −3.35763357775279290676080558542, −2.95421757819640241472299939754, −2.01162949715006552965133117111, −1.04554187861145816688705781311, 0.41900495718309940703905756445, 1.95824817343804162047674388855, 3.20426648088661884531642471624, 4.10649475410977703987132064743, 4.51982871825203050175830658400, 5.12430971758203906334327004529, 5.89135044608834755082190531315, 6.92997781224848026844461888876, 7.964775555137365706437601330589, 8.521402690368147526933329171752

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.311097561\)
\(L(\frac12)\)	\(\approx\)	\(1.311097561\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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