LMFDB - L-function 2-1872-12.11-c1-0-18

Dirichlet series

L(s) = 1

− 1.41i·5^-s + 0.504i·7^-s − 3.16·11^-s + 13^-s + 5.47i·17^-s − 8.44i·19^-s + 0.712·23^-s + 2.99·25^-s − 4.24i·29^-s − 2.96i·31^-s + 0.712·35^-s − 5.74·37^-s − 9.54i·41^-s − 3.46i·43^-s − 12.9·47^-s + ⋯

L(s) = 1

− 0.632i·5^-s + 0.190i·7^-s − 0.953·11^-s + 0.277·13^-s + 1.32i·17^-s − 1.93i·19^-s + 0.148·23^-s + 0.599·25^-s − 0.787i·29^-s − 0.531i·31^-s + 0.120·35^-s − 0.944·37^-s − 1.48i·41^-s − 0.528i·43^-s − 1.89·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1872$ = $2^{4} \cdot 3^{2} \cdot 13$
Sign:	$-0.418 + 0.908i$
Analytic conductor:	$14.9479$
Root analytic conductor:	$3.86626$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1872} (287, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1872,\ (\ :1/2),\ -0.418 + 0.908i)$

Particular Values

$L(1)$	$\approx$	$1.081088342$
$L(\frac12)$	$\approx$	$1.081088342$
$L(\frac{3}{2})$		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 $
	13	$ 1 - T $
good	5	$ 1 + 1.41iT - 5T^{2} $
	7	$ 1 - 0.504iT - 7T^{2} $
	11	$ 1 + 3.16T + 11T^{2} $
	17	$ 1 - 5.47iT - 17T^{2} $
	19	$ 1 + 8.44iT - 19T^{2} $
	23	$ 1 - 0.712T + 23T^{2} $
	29	$ 1 + 4.24iT - 29T^{2} $
	31	$ 1 + 2.96iT - 31T^{2} $
	37	$ 1 + 5.74T + 37T^{2} $
	41	$ 1 + 9.54iT - 41T^{2} $
	43	$ 1 + 3.46iT - 43T^{2} $
	47	$ 1 + 12.9T + 47T^{2} $
	53	$ 1 + 8.30iT - 53T^{2} $
	59	$ 1 + 7.34T + 59T^{2} $
	61	$ 1 - 7.74T + 61T^{2} $
	67	$ 1 - 4.97iT - 67T^{2} $
	71	$ 1 + 1.02T + 71T^{2} $
	73	$ 1 + 9.74T + 73T^{2} $
	79	$ 1 + 7.93iT - 79T^{2} $
	83	$ 1 + 7.34T + 83T^{2} $
	89	$ 1 - 9.54iT - 89T^{2} $
	97	$ 1 + 1.74T + 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

−8.709037339052520537815242663314, −8.494231537180586699436114327336, −7.42779855253820524501948181148, −6.62686647037929914847803560343, −5.61975364682484081873438670682, −4.97669214695906394163000787648, −4.08668614249884787066353327614, −2.93417566780936980139582401814, −1.88959518926525598102624764829, −0.39586876152838468491542788581, 1.42430272825997964298113751405, 2.80011271893989306240606000665, 3.41440422449919638771454757484, 4.66516137410667902768469309259, 5.43936860763251445759935024676, 6.36936759803655462531520299787, 7.16663300389046745624264358855, 7.85054463799691191775073413400, 8.604223775571914117814651664216, 9.619483488354640315752156638497

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.081088342\)
\(L(\frac12)\)	\(\approx\)	\(1.081088342\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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