LMFDB - L-function 2-1800-5.4-c3-0-64

Dirichlet series

L(s) = 1

− 20i·7^-s + 56·11^-s − 86i·13^-s − 106i·17^-s − 4·19^-s − 136i·23^-s − 206·29^-s − 152·31^-s − 282i·37^-s + 246·41^-s + 412i·43^-s + 40i·47^-s − 57·49^-s + 126i·53^-s + 56·59^-s + ⋯

L(s) = 1

− 1.07i·7^-s + 1.53·11^-s − 1.83i·13^-s − 1.51i·17^-s − 0.0482·19^-s − 1.23i·23^-s − 1.31·29^-s − 0.880·31^-s − 1.25i·37^-s + 0.937·41^-s + 1.46i·43^-s + 0.124i·47^-s − 0.166·49^-s + 0.326i·53^-s + 0.123·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1800$ = $2^{3} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.894 + 0.447i$
Analytic conductor:	$106.203$
Root analytic conductor:	$10.3055$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1800} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1800,\ (\ :3/2),\ -0.894 + 0.447i)$

Particular Values

$L(2)$	$\approx$	$1.828973675$
$L(\frac12)$	$\approx$	$1.828973675$
$L(\frac{5}{2})$		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 $
good	7	$ 1 + 20iT - 343T^{2} $
	11	$ 1 - 56T + 1.33e3T^{2} $
	13	$ 1 + 86iT - 2.19e3T^{2} $
	17	$ 1 + 106iT - 4.91e3T^{2} $
	19	$ 1 + 4T + 6.85e3T^{2} $
	23	$ 1 + 136iT - 1.21e4T^{2} $
	29	$ 1 + 206T + 2.43e4T^{2} $
	31	$ 1 + 152T + 2.97e4T^{2} $
	37	$ 1 + 282iT - 5.06e4T^{2} $
	41	$ 1 - 246T + 6.89e4T^{2} $
	43	$ 1 - 412iT - 7.95e4T^{2} $
	47	$ 1 - 40iT - 1.03e5T^{2} $
	53	$ 1 - 126iT - 1.48e5T^{2} $
	59	$ 1 - 56T + 2.05e5T^{2} $
	61	$ 1 + 2T + 2.26e5T^{2} $
	67	$ 1 - 388iT - 3.00e5T^{2} $
	71	$ 1 - 672T + 3.57e5T^{2} $
	73	$ 1 - 1.17e3iT - 3.89e5T^{2} $
	79	$ 1 + 408T + 4.93e5T^{2} $
	83	$ 1 + 668iT - 5.71e5T^{2} $
	89	$ 1 - 66T + 7.04e5T^{2} $
	97	$ 1 - 926iT - 9.12e5T^{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

−8.626442524321218762389049632990, −7.55573606389153586644107481839, −7.20218981909066926692096467040, −6.19987294438630608607436132440, −5.34358387077787987706363247777, −4.30878402071435773728792046289, −3.60952831689328604860708524433, −2.59066943268496010104498614357, −1.10081091824439200030094506926, −0.41546879439459262973680906950, 1.56545409128139241882625076205, 2.00368957554238513808631595433, 3.61744354604618019007755221767, 4.08686649545653370659968502011, 5.30842552944904436132565794151, 6.15658025586462224006549174644, 6.69819097550521283650489719149, 7.66500472705411864483880745688, 8.824768140042083278736390345304, 9.072119065006958770431782414545

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.828973675\)
\(L(\frac12)\)	\(\approx\)	\(1.828973675\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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