LMFDB - L-function 2-2400-40.29-c1-0-4

Dirichlet series

L(s) = 1

− 3^-s + 3.62i·7^-s + 9^-s + 6.20i·11^-s + 0.578·13^-s + 1.42i·17^-s + 5.62i·19^-s − 3.62i·21^-s − 5.62i·23^-s − 27^-s + 2i·29^-s + 2.57·31^-s − 6.20i·33^-s − 7.83·37^-s − 0.578·39^-s + ⋯

L(s) = 1

− 0.577·3^-s + 1.37i·7^-s + 0.333·9^-s + 1.87i·11^-s + 0.160·13^-s + 0.344i·17^-s + 1.29i·19^-s − 0.791i·21^-s − 1.17i·23^-s − 0.192·27^-s + 0.371i·29^-s + 0.463·31^-s − 1.08i·33^-s − 1.28·37^-s − 0.0926·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2400$ = $2^{5} \cdot 3 \cdot 5^{2}$
Sign:	$-0.935 - 0.353i$
Analytic conductor:	$19.1640$
Root analytic conductor:	$4.37768$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2400} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2400,\ (\ :1/2),\ -0.935 - 0.353i)$

Particular Values

$L(1)$	$\approx$	$0.9539871803$
$L(\frac12)$	$\approx$	$0.9539871803$
$L(\frac{3}{2})$		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 + T $
	5	$ 1 $
good	7	$ 1 - 3.62iT - 7T^{2} $
	11	$ 1 - 6.20iT - 11T^{2} $
	13	$ 1 - 0.578T + 13T^{2} $
	17	$ 1 - 1.42iT - 17T^{2} $
	19	$ 1 - 5.62iT - 19T^{2} $
	23	$ 1 + 5.62iT - 23T^{2} $
	29	$ 1 - 2iT - 29T^{2} $
	31	$ 1 - 2.57T + 31T^{2} $
	37	$ 1 + 7.83T + 37T^{2} $
	41	$ 1 - 5.25T + 41T^{2} $
	43	$ 1 + 7.25T + 43T^{2} $
	47	$ 1 + 6.78iT - 47T^{2} $
	53	$ 1 - 2T + 53T^{2} $
	59	$ 1 - 2.20iT - 59T^{2} $
	61	$ 1 + 12.4iT - 61T^{2} $
	67	$ 1 + 4T + 67T^{2} $
	71	$ 1 + 8.41T + 71T^{2} $
	73	$ 1 - 6iT - 73T^{2} $
	79	$ 1 - 5.42T + 79T^{2} $
	83	$ 1 - 3.25T + 83T^{2} $
	89	$ 1 - 13.2T + 89T^{2} $
	97	$ 1 - 4.84iT - 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

−9.317288982282382623501838362649, −8.565817161110790381000909798307, −7.79117738234769431505404394834, −6.82612651431872555189681327329, −6.22285651218383833719852143859, −5.32443562186763852234967095773, −4.73816400500375059151741526361, −3.72293630333557068677002239686, −2.38123385107897960401535928205, −1.66917313509841299267028713508, 0.37447667192830236477776389283, 1.24259243143419065276927415183, 2.96908126547384873592848117038, 3.73551998677796310133921823607, 4.64240621082944485346952258537, 5.52900417118742267813558465146, 6.29724149377622486041225536729, 7.07069383686250477454671984017, 7.72549313057596249835727000512, 8.635705569009749909433863647113

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9539871803\)
\(L(\frac12)\)	\(\approx\)	\(0.9539871803\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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