LMFDB - L-function 2-21e2-21.20-c3-0-34

Dirichlet series

L(s) = 1

− 1.82i·2^-s + 4.68·4^-s + 15.0·5^-s − 23.0i·8^-s − 27.4i·10^-s + 9.89i·11^-s − 67.8i·13^-s − 4.54·16^-s − 70.1·17^-s − 61.4i·19^-s + 70.7·20^-s + 18.0·22^-s − 131. i·23^-s + 102.·25^-s − 123.·26^-s + ⋯

L(s) = 1

− 0.643i·2^-s + 0.585·4^-s + 1.34·5^-s − 1.02i·8^-s − 0.868i·10^-s + 0.271i·11^-s − 1.44i·13^-s − 0.0710·16^-s − 1.00·17^-s − 0.742i·19^-s + 0.790·20^-s + 0.174·22^-s − 1.19i·23^-s + 0.821·25^-s − 0.932·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$441$ = $3^{2} \cdot 7^{2}$
Sign:	$-0.0980 + 0.995i$
Analytic conductor:	$26.0198$
Root analytic conductor:	$5.10096$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{441} (440, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 441,\ (\ :3/2),\ -0.0980 + 0.995i)$

Particular Values

$L(2)$	$\approx$	$2.887808788$
$L(\frac12)$	$\approx$	$2.887808788$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 $
	7	$ 1 $
good	2	$ 1 + 1.82iT - 8T^{2} $
	5	$ 1 - 15.0T + 125T^{2} $
	11	$ 1 - 9.89iT - 1.33e3T^{2} $
	13	$ 1 + 67.8iT - 2.19e3T^{2} $
	17	$ 1 + 70.1T + 4.91e3T^{2} $
	19	$ 1 + 61.4iT - 6.85e3T^{2} $
	23	$ 1 + 131. iT - 1.21e4T^{2} $
	29	$ 1 - 158. iT - 2.43e4T^{2} $
	31	$ 1 - 76.4iT - 2.97e4T^{2} $
	37	$ 1 - 348.T + 5.06e4T^{2} $
	41	$ 1 - 138.T + 6.89e4T^{2} $
	43	$ 1 - 539.T + 7.95e4T^{2} $
	47	$ 1 + 223.T + 1.03e5T^{2} $
	53	$ 1 - 530. iT - 1.48e5T^{2} $
	59	$ 1 + 542.T + 2.05e5T^{2} $
	61	$ 1 + 134. iT - 2.26e5T^{2} $
	67	$ 1 - 320.T + 3.00e5T^{2} $
	71	$ 1 + 416. iT - 3.57e5T^{2} $
	73	$ 1 + 545. iT - 3.89e5T^{2} $
	79	$ 1 + 322.T + 4.93e5T^{2} $
	83	$ 1 - 885.T + 5.71e5T^{2} $
	89	$ 1 + 1.62e3T + 7.04e5T^{2} $
	97	$ 1 - 739. iT - 9.12e5T^{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

−10.70321675767986565440813032407, −9.738926324434174157351830614884, −8.977540225139489844484517002377, −7.62673642274528770675388703216, −6.55043910559509742619228426041, −5.84498142623530173229770753424, −4.59790152067986186448067704217, −2.93686965112647913399963872161, −2.25113243780919595711980064934, −0.916128477039177770364381563902, 1.67379280233725219534066308983, 2.49058399312089437381333562781, 4.28095923822032074205606443372, 5.68203637002544184191914456962, 6.17257876049388053959134137337, 7.04068886314559191958666083226, 8.094917647994419294438118751729, 9.241541962329063674808179931065, 9.815918210410845524649503115187, 11.09083334166926471687574339901

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(2.887808788\)
\(L(\frac12)\)	\(\approx\)	\(2.887808788\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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