LMFDB - L-function 2-300-5.4-c3-0-8

Dirichlet series

L(s) = 1

− 3i·3^-s + 28i·7^-s − 9·9^-s − 24·11^-s − 70i·13^-s − 102i·17^-s − 20·19^-s + 84·21^-s − 72i·23^-s + 27i·27^-s − 306·29^-s − 136·31^-s + 72i·33^-s + 214i·37^-s − 210·39^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 1.51i·7^-s − 0.333·9^-s − 0.657·11^-s − 1.49i·13^-s − 1.45i·17^-s − 0.241·19^-s + 0.872·21^-s − 0.652i·23^-s + 0.192i·27^-s − 1.95·29^-s − 0.787·31^-s + 0.379i·33^-s + 0.950i·37^-s − 0.862·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$300$ = $2^{2} \cdot 3 \cdot 5^{2}$
Sign:	$-0.894 + 0.447i$
Analytic conductor:	$17.7005$
Root analytic conductor:	$4.20720$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{300} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 300,\ (\ :3/2),\ -0.894 + 0.447i)$

Particular Values

$L(2)$	$\approx$	$0.6168154141$
$L(\frac12)$	$\approx$	$0.6168154141$
$L(\frac{5}{2})$		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 + 3iT $
	5	$ 1 $
good	7	$ 1 - 28iT - 343T^{2} $
	11	$ 1 + 24T + 1.33e3T^{2} $
	13	$ 1 + 70iT - 2.19e3T^{2} $
	17	$ 1 + 102iT - 4.91e3T^{2} $
	19	$ 1 + 20T + 6.85e3T^{2} $
	23	$ 1 + 72iT - 1.21e4T^{2} $
	29	$ 1 + 306T + 2.43e4T^{2} $
	31	$ 1 + 136T + 2.97e4T^{2} $
	37	$ 1 - 214iT - 5.06e4T^{2} $
	41	$ 1 + 150T + 6.89e4T^{2} $
	43	$ 1 + 292iT - 7.95e4T^{2} $
	47	$ 1 - 72iT - 1.03e5T^{2} $
	53	$ 1 + 414iT - 1.48e5T^{2} $
	59	$ 1 - 744T + 2.05e5T^{2} $
	61	$ 1 + 418T + 2.26e5T^{2} $
	67	$ 1 + 188iT - 3.00e5T^{2} $
	71	$ 1 - 480T + 3.57e5T^{2} $
	73	$ 1 - 434iT - 3.89e5T^{2} $
	79	$ 1 + 1.35e3T + 4.93e5T^{2} $
	83	$ 1 + 612iT - 5.71e5T^{2} $
	89	$ 1 - 30T + 7.04e5T^{2} $
	97	$ 1 - 286iT - 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

−11.08878124790016621636380164670, −9.904395766119424214776347652034, −8.870127374877108429017804567616, −8.067908766606554171168212463315, −7.07107147013258066738460917740, −5.69885639492656934443017549807, −5.21092384867302336260013302716, −3.12361369486874690740920191813, −2.19196243165657908107263875991, −0.21581774694381423481913778020, 1.73700742106261002077288777173, 3.70522203594383587188523298040, 4.29532029313424573080235283742, 5.68441340119835387532225692903, 6.94549049681046813404907980585, 7.79503360970758461673710478982, 9.014578247177705852645739610895, 9.931593722838556417466438636177, 10.79881034349280661534942936398, 11.34064297322135270274776709491

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.6168154141\)
\(L(\frac12)\)	\(\approx\)	\(0.6168154141\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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