LMFDB - L-function 2-600-5.4-c7-0-38

Dirichlet series

L(s) = 1

− 27i·3^-s + 120i·7^-s − 729·9^-s − 7.19e3·11^-s + 9.62e3i·13^-s + 1.86e4i·17^-s − 7.00e3·19^-s + 3.24e3·21^-s + 6.37e4i·23^-s + 1.96e4i·27^-s − 2.93e4·29^-s + 8.79e4·31^-s + 1.94e5i·33^-s + 2.27e5i·37^-s + 2.59e5·39^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.132i·7^-s − 0.333·9^-s − 1.63·11^-s + 1.21i·13^-s + 0.921i·17^-s − 0.234·19^-s + 0.0763·21^-s + 1.09i·23^-s + 0.192i·27^-s − 0.223·29^-s + 0.530·31^-s + 0.941i·33^-s + 0.739i·37^-s + 0.701·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$600$ = $2^{3} \cdot 3 \cdot 5^{2}$
Sign:	$-0.447 + 0.894i$
Analytic conductor:	$187.431$
Root analytic conductor:	$13.6905$
Motivic weight:	$7$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{600} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 600,\ (\ :7/2),\ -0.447 + 0.894i)$

Particular Values

$L(4)$	$\approx$	$0.5345314944$
$L(\frac12)$	$\approx$	$0.5345314944$
$L(\frac{9}{2})$		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 $
	3	$ 1 + 27iT $
	5	$ 1 $
good	7	$ 1 - 120iT - 8.23e5T^{2} $
	11	$ 1 + 7.19e3T + 1.94e7T^{2} $
	13	$ 1 - 9.62e3iT - 6.27e7T^{2} $
	17	$ 1 - 1.86e4iT - 4.10e8T^{2} $
	19	$ 1 + 7.00e3T + 8.93e8T^{2} $
	23	$ 1 - 6.37e4iT - 3.40e9T^{2} $
	29	$ 1 + 2.93e4T + 1.72e10T^{2} $
	31	$ 1 - 8.79e4T + 2.75e10T^{2} $
	37	$ 1 - 2.27e5iT - 9.49e10T^{2} $
	41	$ 1 + 1.60e5T + 1.94e11T^{2} $
	43	$ 1 + 1.36e5iT - 2.71e11T^{2} $
	47	$ 1 + 1.20e6iT - 5.06e11T^{2} $
	53	$ 1 - 3.98e5iT - 1.17e12T^{2} $
	59	$ 1 + 1.15e6T + 2.48e12T^{2} $
	61	$ 1 + 2.07e6T + 3.14e12T^{2} $
	67	$ 1 + 4.07e6iT - 6.06e12T^{2} $
	71	$ 1 + 3.83e5T + 9.09e12T^{2} $
	73	$ 1 + 3.00e6iT - 1.10e13T^{2} $
	79	$ 1 - 4.94e6T + 1.92e13T^{2} $
	83	$ 1 - 9.16e6iT - 2.71e13T^{2} $
	89	$ 1 + 7.30e6T + 4.42e13T^{2} $
	97	$ 1 + 6.90e5iT - 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

−9.164778839103739483077602397520, −8.258384559084609685099696772484, −7.55577006921015380535620999451, −6.61015434050889827479681354916, −5.69352699490905185809490309815, −4.76335132244669478297765870742, −3.52092370436999700619546234307, −2.35712715046603323715863161325, −1.55860369858555800863079413818, −0.12876054469593669975159679996, 0.71864077175184878371724231034, 2.48125264319440931423656543434, 3.08306618528699706637684322966, 4.44656836656036893678837131322, 5.20749898190985873271639391715, 6.03149308141824495209859085997, 7.38310155675204498438207364133, 8.050082635232074792127757616768, 8.958484639219122782652552273585, 10.06328982964450447432512800091

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(0.5345314944\)
\(L(\frac12)\)	\(\approx\)	\(0.5345314944\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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