LMFDB - L-function 2-20e2-5.4-c9-0-79

Dirichlet series

L(s) = 1

− 210. i·3^-s − 9.90e3i·7^-s − 2.44e4·9^-s − 3.64e4·11^-s − 1.64e5i·13^-s + 8.23e4i·17^-s − 6.09e5·19^-s − 2.08e6·21^-s + 1.88e6i·23^-s + 1.01e6i·27^-s − 3.39e5·29^-s − 5.47e5·31^-s + 7.66e6i·33^-s + 5.25e6i·37^-s − 3.46e7·39^-s + ⋯

L(s) = 1

− 1.49i·3^-s − 1.55i·7^-s − 1.24·9^-s − 0.750·11^-s − 1.60i·13^-s + 0.239i·17^-s − 1.07·19^-s − 2.33·21^-s + 1.40i·23^-s + 0.365i·27^-s − 0.0890·29^-s − 0.106·31^-s + 1.12i·33^-s + 0.461i·37^-s − 2.39·39^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$400$ = $2^{4} \cdot 5^{2}$
Sign:	$0.894 - 0.447i$
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:	$(2,\ 400,\ (\ :9/2),\ 0.894 - 0.447i)$

Particular Values

$L(5)$	$\approx$	$0.1522017698$
$L(\frac12)$	$\approx$	$0.1522017698$
$L(\frac{11}{2})$		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 + 210. iT - 1.96e4T^{2} $
	7	$ 1 + 9.90e3iT - 4.03e7T^{2} $
	11	$ 1 + 3.64e4T + 2.35e9T^{2} $
	13	$ 1 + 1.64e5iT - 1.06e10T^{2} $
	17	$ 1 - 8.23e4iT - 1.18e11T^{2} $
	19	$ 1 + 6.09e5T + 3.22e11T^{2} $
	23	$ 1 - 1.88e6iT - 1.80e12T^{2} $
	29	$ 1 + 3.39e5T + 1.45e13T^{2} $
	31	$ 1 + 5.47e5T + 2.64e13T^{2} $
	37	$ 1 - 5.25e6iT - 1.29e14T^{2} $
	41	$ 1 - 2.05e6T + 3.27e14T^{2} $
	43	$ 1 + 6.76e6iT - 5.02e14T^{2} $
	47	$ 1 + 3.15e7iT - 1.11e15T^{2} $
	53	$ 1 - 4.89e7iT - 3.29e15T^{2} $
	59	$ 1 - 8.77e7T + 8.66e15T^{2} $
	61	$ 1 - 3.84e7T + 1.16e16T^{2} $
	67	$ 1 - 1.36e8iT - 2.72e16T^{2} $
	71	$ 1 + 3.49e8T + 4.58e16T^{2} $
	73	$ 1 + 1.61e8iT - 5.88e16T^{2} $
	79	$ 1 + 1.26e8T + 1.19e17T^{2} $
	83	$ 1 + 2.87e8iT - 1.86e17T^{2} $
	89	$ 1 - 5.63e8T + 3.50e17T^{2} $
	97	$ 1 + 4.71e8iT - 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

−8.512202469086879843914731719772, −7.66869929870064254366112796351, −7.33480625647035220985531196299, −6.28501547042185576762236530694, −5.30420056927059400361793881496, −3.90241198539099715163165138224, −2.80665208883509954974571775726, −1.61621447037682456041063898463, −0.75960086684049779893969625301, −0.03369591103947671414233440183, 2.04894280587728252100971859916, 2.79742379647505793393120930995, 4.12284884774896575050444350530, 4.80200198833781222605847069419, 5.70550377896811179424707004515, 6.68067740852695020333439663039, 8.315696454191997221130591905324, 8.965285417480988709504493815341, 9.566952521426955954255183252111, 10.54412632349062632767403007183

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 $\F_{q}$

\(L(5)\)	\(\approx\)	\(0.1522017698\)
\(L(\frac12)\)	\(\approx\)	\(0.1522017698\)
\(L(\frac{11}{2})\)		not available
\(L(1)\)		not available

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