LMFDB - L-function 2-2736-76.75-c1-0-2

Dirichlet series

L(s) = 1

− 3·5^-s + 1.73i·7^-s + 5.19i·11^-s + 6.92i·13^-s − 3·17^-s + (4 − 1.73i)19^-s − 3.46i·23^-s + 4·25^-s − 4·31^-s − 5.19i·35^-s − 6.92i·37^-s + 6.92i·41^-s + 8.66i·43^-s − 8.66i·47^-s + 4·49^-s + ⋯

L(s) = 1

− 1.34·5^-s + 0.654i·7^-s + 1.56i·11^-s + 1.92i·13^-s − 0.727·17^-s + (0.917 − 0.397i)19^-s − 0.722i·23^-s + 0.800·25^-s − 0.718·31^-s − 0.878i·35^-s − 1.13i·37^-s + 1.08i·41^-s + 1.32i·43^-s − 1.26i·47^-s + 0.571·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$-0.917 + 0.397i$
Analytic conductor:	$21.8470$
Root analytic conductor:	$4.67408$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2736} (2431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2736,\ (\ :1/2),\ -0.917 + 0.397i)$

Particular Values

$L(1)$	$\approx$	$0.4034990806$
$L(\frac12)$	$\approx$	$0.4034990806$
$L(\frac{3}{2})$		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 $
	19	$ 1 + (-4 + 1.73i)T $
good	5	$ 1 + 3T + 5T^{2} $
	7	$ 1 - 1.73iT - 7T^{2} $
	11	$ 1 - 5.19iT - 11T^{2} $
	13	$ 1 - 6.92iT - 13T^{2} $
	17	$ 1 + 3T + 17T^{2} $
	23	$ 1 + 3.46iT - 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 + 4T + 31T^{2} $
	37	$ 1 + 6.92iT - 37T^{2} $
	41	$ 1 - 6.92iT - 41T^{2} $
	43	$ 1 - 8.66iT - 43T^{2} $
	47	$ 1 + 8.66iT - 47T^{2} $
	53	$ 1 + 6.92iT - 53T^{2} $
	59	$ 1 + 12T + 59T^{2} $
	61	$ 1 - 7T + 61T^{2} $
	67	$ 1 - 8T + 67T^{2} $
	71	$ 1 + 12T + 71T^{2} $
	73	$ 1 + 5T + 73T^{2} $
	79	$ 1 + 8T + 79T^{2} $
	83	$ 1 - 3.46iT - 83T^{2} $
	89	$ 1 - 6.92iT - 89T^{2} $
	97	$ 1 + 6.92iT - 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

−9.212291945496614031425153989737, −8.558571977932800597808796766576, −7.60665540134120069815031270274, −7.05976695935073563409145620430, −6.46924621122733409412709229248, −5.13698122377714035576864543291, −4.41426893708808648998914503636, −3.93691725114823921101828932161, −2.62321827240139958344867851899, −1.72354329503954401126436974481, 0.15325068011453699940967475640, 1.05701477312683819753789372682, 3.01558305608033511446890935733, 3.44885171456428312088057428451, 4.24188192936261500795265067956, 5.38686810365043352245533050439, 5.93724806769214772529088030741, 7.20448437257015446406325909788, 7.64507147202340724519181813769, 8.288286964029819327067385635200

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.4034990806\)
\(L(\frac12)\)	\(\approx\)	\(0.4034990806\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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