LMFDB - L-function 2-384-8.5-c3-0-13

Dirichlet series

L(s) = 1

− 3i·3^-s + 8i·5^-s − 12·7^-s − 9·9^-s − 12i·11^-s + 20i·13^-s + 24·15^-s + 62·17^-s − 108i·19^-s + 36i·21^-s + 72·23^-s + 61·25^-s + 27i·27^-s − 128i·29^-s + 204·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.715i·5^-s − 0.647·7^-s − 0.333·9^-s − 0.328i·11^-s + 0.426i·13^-s + 0.413·15^-s + 0.884·17^-s − 1.30i·19^-s + 0.374i·21^-s + 0.652·23^-s + 0.487·25^-s + 0.192i·27^-s − 0.819i·29^-s + 1.18·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$384$ = $2^{7} \cdot 3$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$22.6567$
Root analytic conductor:	$4.75990$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{384} (193, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 384,\ (\ :3/2),\ 0.707 + 0.707i)$

Particular Values

$L(2)$	$\approx$	$1.662048924$
$L(\frac12)$	$\approx$	$1.662048924$
$L(\frac{5}{2})$		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 $
good	5	$ 1 - 8iT - 125T^{2} $
	7	$ 1 + 12T + 343T^{2} $
	11	$ 1 + 12iT - 1.33e3T^{2} $
	13	$ 1 - 20iT - 2.19e3T^{2} $
	17	$ 1 - 62T + 4.91e3T^{2} $
	19	$ 1 + 108iT - 6.85e3T^{2} $
	23	$ 1 - 72T + 1.21e4T^{2} $
	29	$ 1 + 128iT - 2.43e4T^{2} $
	31	$ 1 - 204T + 2.97e4T^{2} $
	37	$ 1 - 228iT - 5.06e4T^{2} $
	41	$ 1 + 22T + 6.89e4T^{2} $
	43	$ 1 + 204iT - 7.95e4T^{2} $
	47	$ 1 - 600T + 1.03e5T^{2} $
	53	$ 1 + 256iT - 1.48e5T^{2} $
	59	$ 1 + 828iT - 2.05e5T^{2} $
	61	$ 1 + 84iT - 2.26e5T^{2} $
	67	$ 1 + 348iT - 3.00e5T^{2} $
	71	$ 1 + 456T + 3.57e5T^{2} $
	73	$ 1 - 822T + 3.89e5T^{2} $
	79	$ 1 - 1.35e3T + 4.93e5T^{2} $
	83	$ 1 + 108iT - 5.71e5T^{2} $
	89	$ 1 + 938T + 7.04e5T^{2} $
	97	$ 1 - 1.27e3T + 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

−10.88758045829862777463725896634, −9.908757253579809690057150085985, −8.970381761779671909165505777745, −7.88366026597079824288528619576, −6.84660943388273585944637826379, −6.33506803100283482008421661977, −4.99321205682029885702958347931, −3.41710824678293383153661424220, −2.50033972193491986500237740914, −0.72028353492479047861093909844, 1.04797873822628727908765330350, 2.92198359829609070233142221255, 4.04303773165259767684355044715, 5.16738416771866755775375752886, 6.03029862734700909799207496381, 7.36200899198744230967914767662, 8.405451713842020603646094967365, 9.278962047227124056166872836397, 10.07581722560803102455752514939, 10.80794673550282631160108035630

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.662048924\)
\(L(\frac12)\)	\(\approx\)	\(1.662048924\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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