LMFDB - L-function 2-1216-152.37-c2-0-18

Dirichlet series

L(s) = 1

− 5.07·3^-s + 5.51i·5^-s − 8.74·7^-s + 16.7·9^-s + 0.103i·11^-s − 7.16·13^-s − 28.0i·15^-s + 24.4·17^-s + (7.40 + 17.4i)19^-s + 44.3·21^-s + 10.9·23^-s − 5.44·25^-s − 39.3·27^-s + 12.9·29^-s − 40.6i·31^-s + ⋯

L(s) = 1

− 1.69·3^-s + 1.10i·5^-s − 1.24·7^-s + 1.86·9^-s + 0.00938i·11^-s − 0.551·13^-s − 1.86i·15^-s + 1.43·17^-s + (0.389 + 0.920i)19^-s + 2.11·21^-s + 0.473·23^-s − 0.217·25^-s − 1.45·27^-s + 0.447·29^-s − 1.31i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1216$ = $2^{6} \cdot 19$
Sign:	$-0.615 - 0.788i$
Analytic conductor:	$33.1336$
Root analytic conductor:	$5.75617$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1216} (417, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1216,\ (\ :1),\ -0.615 - 0.788i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.6392165536$
$L(\frac12)$	$\approx$	$0.6392165536$
$L(2)$		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 $
	19	$ 1 + (-7.40 - 17.4i)T $
good	3	$ 1 + 5.07T + 9T^{2} $
	5	$ 1 - 5.51iT - 25T^{2} $
	7	$ 1 + 8.74T + 49T^{2} $
	11	$ 1 - 0.103iT - 121T^{2} $
	13	$ 1 + 7.16T + 169T^{2} $
	17	$ 1 - 24.4T + 289T^{2} $
	23	$ 1 - 10.9T + 529T^{2} $
	29	$ 1 - 12.9T + 841T^{2} $
	31	$ 1 + 40.6iT - 961T^{2} $
	37	$ 1 - 42.1T + 1.36e3T^{2} $
	41	$ 1 - 1.13iT - 1.68e3T^{2} $
	43	$ 1 - 63.9iT - 1.84e3T^{2} $
	47	$ 1 - 17.7T + 2.20e3T^{2} $
	53	$ 1 + 52.4T + 2.80e3T^{2} $
	59	$ 1 - 117.T + 3.48e3T^{2} $
	61	$ 1 - 26.0iT - 3.72e3T^{2} $
	67	$ 1 + 67.7T + 4.48e3T^{2} $
	71	$ 1 + 92.2iT - 5.04e3T^{2} $
	73	$ 1 + 20.5T + 5.32e3T^{2} $
	79	$ 1 + 72.3iT - 6.24e3T^{2} $
	83	$ 1 - 69.7iT - 6.88e3T^{2} $
	89	$ 1 - 8.48iT - 7.92e3T^{2} $
	97	$ 1 - 130. iT - 9.40e3T^{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

−9.952568299599059375030853900350, −9.582423471388336797470606719292, −7.78961312777540603182376073719, −7.15072855780694234698704026951, −6.22505226097902381068727853303, −5.99084542793399691298392464908, −4.92235248712665351827834947827, −3.70297747013631533013200112774, −2.76351237177504501530573290626, −0.940308500126287649781939654561, 0.36225387506820272361548251137, 1.13394461264813053043276296021, 3.05771716963374284962454719415, 4.36159273018383303686951463905, 5.22009545903227097237958327065, 5.65523656189071335277456543151, 6.69584633185982918783398909838, 7.25458402086808077374602792156, 8.553930831808940287093213850147, 9.532620372677067885673567118765

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.6392165536\)
\(L(\frac12)\)	\(\approx\)	\(0.6392165536\)
\(L(2)\)		not available
\(L(1)\)		not available

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