LMFDB - L-function 2-8820-21.20-c1-0-14

Dirichlet series

L(s) = 1

− 5^-s + 2.99i·11^-s − 3.03i·13^-s + 2.51·17^-s − 0.335i·19^-s + 1.54i·23^-s + 25^-s − 0.257i·29^-s + 5.97i·31^-s − 3.77·37^-s + 11.1·41^-s − 9.27·43^-s + 0.877·47^-s − 5.23i·53^-s − 2.99i·55^-s + ⋯

L(s) = 1

− 0.447·5^-s + 0.903i·11^-s − 0.842i·13^-s + 0.610·17^-s − 0.0770i·19^-s + 0.322i·23^-s + 0.200·25^-s − 0.0477i·29^-s + 1.07i·31^-s − 0.619·37^-s + 1.74·41^-s − 1.41·43^-s + 0.127·47^-s − 0.719i·53^-s − 0.404i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$8820$ = $2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2}$
Sign:	$0.192 - 0.981i$
Analytic conductor:	$70.4280$
Root analytic conductor:	$8.39214$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{8820} (881, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 8820,\ (\ :1/2),\ 0.192 - 0.981i)$

Particular Values

$L(1)$	$\approx$	$1.383458677$
$L(\frac12)$	$\approx$	$1.383458677$
$L(\frac{3}{2})$		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 $
	5	$ 1 + T $
	7	$ 1 $
good	11	$ 1 - 2.99iT - 11T^{2} $
	13	$ 1 + 3.03iT - 13T^{2} $
	17	$ 1 - 2.51T + 17T^{2} $
	19	$ 1 + 0.335iT - 19T^{2} $
	23	$ 1 - 1.54iT - 23T^{2} $
	29	$ 1 + 0.257iT - 29T^{2} $
	31	$ 1 - 5.97iT - 31T^{2} $
	37	$ 1 + 3.77T + 37T^{2} $
	41	$ 1 - 11.1T + 41T^{2} $
	43	$ 1 + 9.27T + 43T^{2} $
	47	$ 1 - 0.877T + 47T^{2} $
	53	$ 1 + 5.23iT - 53T^{2} $
	59	$ 1 - 3.88T + 59T^{2} $
	61	$ 1 + 2.67iT - 61T^{2} $
	67	$ 1 - 1.10T + 67T^{2} $
	71	$ 1 - 3.70iT - 71T^{2} $
	73	$ 1 - 0.740iT - 73T^{2} $
	79	$ 1 + 8.67T + 79T^{2} $
	83	$ 1 + 6.28T + 83T^{2} $
	89	$ 1 + 4.99T + 89T^{2} $
	97	$ 1 - 9.11iT - 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

−7.88073238722006233830710916908, −7.23947407794198469056928894132, −6.70523343617599506966796146137, −5.72832300633418069961122383515, −5.14660288363347921266349241463, −4.43124172121192185015682249016, −3.58543146601920862847945092558, −2.94040890196022564587630354407, −1.92742207878696419777671447214, −0.907061268751395042823396018084, 0.38666890158088665356456198093, 1.46047330741662008492655248583, 2.52893268182303862153798547054, 3.34756990650657258041563799903, 4.05879768905862212475660874260, 4.71458813307536579016881023404, 5.67621045052858167757317026922, 6.14793876151865970726972281456, 7.01355401269276622399484379367, 7.57451913101835929722432675091

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.383458677\)
\(L(\frac12)\)	\(\approx\)	\(1.383458677\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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