LMFDB - L-function 2-2300-5.4-c3-0-92

Dirichlet series

L(s) = 1

− 3.72i·3^-s − 11.8i·7^-s + 13.1·9^-s + 64.0·11^-s − 77.5i·13^-s − 25.6i·17^-s − 51.4·19^-s − 44.2·21^-s − 23i·23^-s − 149. i·27^-s − 68.3·29^-s − 67.5·31^-s − 238. i·33^-s + 149. i·37^-s − 289.·39^-s + ⋯

L(s) = 1

− 0.717i·3^-s − 0.640i·7^-s + 0.485·9^-s + 1.75·11^-s − 1.65i·13^-s − 0.365i·17^-s − 0.621·19^-s − 0.459·21^-s − 0.208i·23^-s − 1.06i·27^-s − 0.437·29^-s − 0.391·31^-s − 1.25i·33^-s + 0.663i·37^-s − 1.18·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2300$ = $2^{2} \cdot 5^{2} \cdot 23$
Sign:	$-0.894 + 0.447i$
Analytic conductor:	$135.704$
Root analytic conductor:	$11.6492$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2300} (1749, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2300,\ (\ :3/2),\ -0.894 + 0.447i)$

Particular Values

$L(2)$	$\approx$	$2.328425761$
$L(\frac12)$	$\approx$	$2.328425761$
$L(\frac{5}{2})$		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 $
	5	$ 1 $
	23	$ 1 + 23iT $
good	3	$ 1 + 3.72iT - 27T^{2} $
	7	$ 1 + 11.8iT - 343T^{2} $
	11	$ 1 - 64.0T + 1.33e3T^{2} $
	13	$ 1 + 77.5iT - 2.19e3T^{2} $
	17	$ 1 + 25.6iT - 4.91e3T^{2} $
	19	$ 1 + 51.4T + 6.85e3T^{2} $
	29	$ 1 + 68.3T + 2.43e4T^{2} $
	31	$ 1 + 67.5T + 2.97e4T^{2} $
	37	$ 1 - 149. iT - 5.06e4T^{2} $
	41	$ 1 + 249.T + 6.89e4T^{2} $
	43	$ 1 + 534. iT - 7.95e4T^{2} $
	47	$ 1 + 141. iT - 1.03e5T^{2} $
	53	$ 1 + 76.1iT - 1.48e5T^{2} $
	59	$ 1 - 816.T + 2.05e5T^{2} $
	61	$ 1 - 356.T + 2.26e5T^{2} $
	67	$ 1 - 671. iT - 3.00e5T^{2} $
	71	$ 1 + 746.T + 3.57e5T^{2} $
	73	$ 1 + 162. iT - 3.89e5T^{2} $
	79	$ 1 + 194.T + 4.93e5T^{2} $
	83	$ 1 - 246. iT - 5.71e5T^{2} $
	89	$ 1 - 836.T + 7.04e5T^{2} $
	97	$ 1 - 53.5iT - 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

−8.324292762527581658270179681576, −7.31154646639522747100259453863, −6.97100988736231759990207864640, −6.15747975396527340544216983803, −5.22894708027250225824981368821, −4.10449347550952274283020017059, −3.52531663484988067976719818087, −2.20551138565150544937803227205, −1.19189589897306386792130280522, −0.49007294340007816644173380044, 1.35597416554001898031635972874, 2.08285276257850498832819048693, 3.57538025633753915597389087542, 4.13387697914081521124419218496, 4.78307977366567560683432303497, 5.96042837184958976218713982777, 6.60997339055340627759755384328, 7.27949649299263692400550842520, 8.542472562055336487857188308962, 9.160014579712816931644760474041

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(2.328425761\)
\(L(\frac12)\)	\(\approx\)	\(2.328425761\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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