LMFDB - L-function 2-42e2-21.20-c3-0-18

Dirichlet series

L(s) = 1

− 8.73·5^-s − 8.78i·11^-s − 11.8i·13^-s + 44.5·17^-s − 11.6i·19^-s + 142. i·23^-s − 48.6·25^-s − 234. i·29^-s + 291. i·31^-s − 88.9·37^-s − 145.·41^-s + 144.·43^-s − 240.·47^-s − 304. i·53^-s + 76.7i·55^-s + ⋯

L(s) = 1

− 0.781·5^-s − 0.240i·11^-s − 0.252i·13^-s + 0.636·17^-s − 0.140i·19^-s + 1.29i·23^-s − 0.389·25^-s − 1.49i·29^-s + 1.69i·31^-s − 0.395·37^-s − 0.555·41^-s + 0.512·43^-s − 0.745·47^-s − 0.788i·53^-s + 0.188i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1764$ = $2^{2} \cdot 3^{2} \cdot 7^{2}$
Sign:	$0.970 + 0.239i$
Analytic conductor:	$104.079$
Root analytic conductor:	$10.2019$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1764} (881, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1764,\ (\ :3/2),\ 0.970 + 0.239i)$

Particular Values

$L(2)$	$\approx$	$1.462720989$
$L(\frac12)$	$\approx$	$1.462720989$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	7	$ 1 $
good	5	$ 1 + 8.73T + 125T^{2} $
	11	$ 1 + 8.78iT - 1.33e3T^{2} $
	13	$ 1 + 11.8iT - 2.19e3T^{2} $
	17	$ 1 - 44.5T + 4.91e3T^{2} $
	19	$ 1 + 11.6iT - 6.85e3T^{2} $
	23	$ 1 - 142. iT - 1.21e4T^{2} $
	29	$ 1 + 234. iT - 2.43e4T^{2} $
	31	$ 1 - 291. iT - 2.97e4T^{2} $
	37	$ 1 + 88.9T + 5.06e4T^{2} $
	41	$ 1 + 145.T + 6.89e4T^{2} $
	43	$ 1 - 144.T + 7.95e4T^{2} $
	47	$ 1 + 240.T + 1.03e5T^{2} $
	53	$ 1 + 304. iT - 1.48e5T^{2} $
	59	$ 1 - 7.08T + 2.05e5T^{2} $
	61	$ 1 - 172. iT - 2.26e5T^{2} $
	67	$ 1 + 486.T + 3.00e5T^{2} $
	71	$ 1 + 653. iT - 3.57e5T^{2} $
	73	$ 1 + 114. iT - 3.89e5T^{2} $
	79	$ 1 - 294.T + 4.93e5T^{2} $
	83	$ 1 - 877.T + 5.71e5T^{2} $
	89	$ 1 - 1.42e3T + 7.04e5T^{2} $
	97	$ 1 - 738. 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

−8.838031064472930290494027809973, −7.975280255282774932480034807301, −7.53305415976887220553811430507, −6.55066701413550580425960623564, −5.63116533868870755726596312337, −4.80989862254546904009744375712, −3.74430554676862918668766607791, −3.14449625631485910000375838747, −1.76345992364009419858013169138, −0.53241168227028838654452383535, 0.60819333770907363965458199966, 1.91515047850102293590651896082, 3.09076443165906947149595189370, 3.98847821606779540826409143092, 4.74511484897290071237287946585, 5.74997409790113609339389686215, 6.66497084898448669827523658462, 7.47291537939674484589831381631, 8.113110278471278230389800999832, 8.896098842465585154172033115150

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.462720989\)
\(L(\frac12)\)	\(\approx\)	\(1.462720989\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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