LMFDB - L-function 2-20e2-5.4-c7-0-2

Dirichlet series

L(s) = 1

− 56.2i·3^-s + 1.34e3i·7^-s − 978.·9^-s − 4.38e3·11^-s − 3.76e3i·13^-s − 3.84e3i·17^-s + 3.89e4·19^-s + 7.56e4·21^-s − 5.93e4i·23^-s − 6.80e4i·27^-s − 2.07e5·29^-s + 2.86e5·31^-s + 2.46e5i·33^-s + 4.53e5i·37^-s − 2.11e5·39^-s + ⋯

L(s) = 1

− 1.20i·3^-s + 1.48i·7^-s − 0.447·9^-s − 0.992·11^-s − 0.474i·13^-s − 0.189i·17^-s + 1.30·19^-s + 1.78·21^-s − 1.01i·23^-s − 0.665i·27^-s − 1.58·29^-s + 1.72·31^-s + 1.19i·33^-s + 1.47i·37^-s − 0.571·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$400$ = $2^{4} \cdot 5^{2}$
Sign:	$-0.447 - 0.894i$
Analytic conductor:	$124.954$
Root analytic conductor:	$11.1782$
Motivic weight:	$7$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{400} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 400,\ (\ :7/2),\ -0.447 - 0.894i)$

Particular Values

$L(4)$	$\approx$	$0.3397753111$
$L(\frac12)$	$\approx$	$0.3397753111$
$L(\frac{9}{2})$		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 + 56.2iT - 2.18e3T^{2} $
	7	$ 1 - 1.34e3iT - 8.23e5T^{2} $
	11	$ 1 + 4.38e3T + 1.94e7T^{2} $
	13	$ 1 + 3.76e3iT - 6.27e7T^{2} $
	17	$ 1 + 3.84e3iT - 4.10e8T^{2} $
	19	$ 1 - 3.89e4T + 8.93e8T^{2} $
	23	$ 1 + 5.93e4iT - 3.40e9T^{2} $
	29	$ 1 + 2.07e5T + 1.72e10T^{2} $
	31	$ 1 - 2.86e5T + 2.75e10T^{2} $
	37	$ 1 - 4.53e5iT - 9.49e10T^{2} $
	41	$ 1 - 5.98e3T + 1.94e11T^{2} $
	43	$ 1 - 2.16e5iT - 2.71e11T^{2} $
	47	$ 1 + 4.06e5iT - 5.06e11T^{2} $
	53	$ 1 + 1.38e6iT - 1.17e12T^{2} $
	59	$ 1 + 1.88e6T + 2.48e12T^{2} $
	61	$ 1 + 2.77e6T + 3.14e12T^{2} $
	67	$ 1 - 3.24e6iT - 6.06e12T^{2} $
	71	$ 1 - 5.71e5T + 9.09e12T^{2} $
	73	$ 1 - 2.07e6iT - 1.10e13T^{2} $
	79	$ 1 + 1.92e6T + 1.92e13T^{2} $
	83	$ 1 + 6.83e6iT - 2.71e13T^{2} $
	89	$ 1 - 3.55e6T + 4.42e13T^{2} $
	97	$ 1 - 1.26e7iT - 8.07e13T^{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

−10.36528522364772092667658966590, −9.379105073349822555100580715234, −8.272996907387343781141807921514, −7.75908654678905211624470950525, −6.63524631899505417385615889599, −5.75362156040322564552731669906, −4.91868995610243790616437317865, −2.99473172098501877700142538240, −2.33191668641751950209061584780, −1.16680528282857103256936404780, 0.07161218129733511986946596947, 1.40035725565120737257895835614, 3.11240596840617704522062351376, 3.98002778237061830644907178972, 4.74913520093527326904052626082, 5.75086110306128783361999630700, 7.25645406161879112695156568025, 7.77298594328095683872344014456, 9.281150819844419529388795699164, 9.809991120127635153621019910389

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 $\F_{q}$
Belyi maps

\(L(4)\)	\(\approx\)	\(0.3397753111\)
\(L(\frac12)\)	\(\approx\)	\(0.3397753111\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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