LMFDB - L-function 2-20e2-5.4-c9-0-70

Dirichlet series

L(s) = 1

− 174i·3^-s + 4.65e3i·7^-s − 1.05e4·9^-s − 2.89e4·11^-s − 1.64e5i·13^-s + 5.94e5i·17^-s − 2.95e5·19^-s + 8.10e5·21^-s − 2.54e6i·23^-s − 1.58e6i·27^-s + 3.72e6·29^-s − 2.33e6·31^-s + 5.04e6i·33^-s − 1.08e7i·37^-s − 2.86e7·39^-s + ⋯

L(s) = 1

− 1.24i·3^-s + 0.733i·7^-s − 0.538·9^-s − 0.597·11^-s − 1.59i·13^-s + 1.72i·17^-s − 0.520·19^-s + 0.909·21^-s − 1.89i·23^-s − 0.572i·27^-s + 0.977·29^-s − 0.454·31^-s + 0.740i·33^-s − 0.950i·37^-s − 1.98·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(10-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$400$ = $2^{4} \cdot 5^{2}$
Sign:	$-0.447 - 0.894i$
Analytic conductor:	$206.014$
Root analytic conductor:	$14.3531$
Motivic weight:	$9$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{400} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 400,\ (\ :9/2),\ -0.447 - 0.894i)$

Particular Values

$L(5)$	$\approx$	$0.3137003161$
$L(\frac12)$	$\approx$	$0.3137003161$
$L(\frac{11}{2})$		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 $
	5	$ 1 $
good	3	$ 1 + 174iT - 1.96e4T^{2} $
	7	$ 1 - 4.65e3iT - 4.03e7T^{2} $
	11	$ 1 + 2.89e4T + 2.35e9T^{2} $
	13	$ 1 + 1.64e5iT - 1.06e10T^{2} $
	17	$ 1 - 5.94e5iT - 1.18e11T^{2} $
	19	$ 1 + 2.95e5T + 3.22e11T^{2} $
	23	$ 1 + 2.54e6iT - 1.80e12T^{2} $
	29	$ 1 - 3.72e6T + 1.45e13T^{2} $
	31	$ 1 + 2.33e6T + 2.64e13T^{2} $
	37	$ 1 + 1.08e7iT - 1.29e14T^{2} $
	41	$ 1 - 2.15e7T + 3.27e14T^{2} $
	43	$ 1 + 1.08e7iT - 5.02e14T^{2} $
	47	$ 1 - 5.17e6iT - 1.11e15T^{2} $
	53	$ 1 - 9.81e7iT - 3.29e15T^{2} $
	59	$ 1 - 1.61e7T + 8.66e15T^{2} $
	61	$ 1 + 4.39e7T + 1.16e16T^{2} $
	67	$ 1 + 8.15e7iT - 2.72e16T^{2} $
	71	$ 1 + 1.61e8T + 4.58e16T^{2} $
	73	$ 1 + 2.47e8iT - 5.88e16T^{2} $
	79	$ 1 + 5.83e8T + 1.19e17T^{2} $
	83	$ 1 - 1.45e7iT - 1.86e17T^{2} $
	89	$ 1 + 4.70e8T + 3.50e17T^{2} $
	97	$ 1 - 1.17e8iT - 7.60e17T^{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

−8.739409758580479119354466602029, −8.190370531527920109481723557088, −7.40186343396052309944558012659, −6.20120571221971086136761349188, −5.75090440756367043104765541073, −4.34415588405956701059091691749, −2.83615054466080691516621453998, −2.15702874149162017771939641968, −0.994341661542889499612272260595, −0.06092649407202144093455575468, 1.34312073484446078547950898314, 2.74643954238966933276088115537, 3.85631710785207380048170639180, 4.56612791953594918913883682536, 5.35417571632987187600244137472, 6.78771128178437494259490083358, 7.53529534913421588707781736533, 8.878152589881716283060073796955, 9.610122505627921106658237329307, 10.17022110394357071476635254281

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 $\F_{q}$

\(L(5)\)	\(\approx\)	\(0.3137003161\)
\(L(\frac12)\)	\(\approx\)	\(0.3137003161\)
\(L(\frac{11}{2})\)		not available
\(L(1)\)		not available

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