LMFDB - L-function 2-384-8.5-c9-0-36

Dirichlet series

L(s) = 1

+ 81i·3^-s − 2.17e3i·5^-s − 7.65e3·7^-s − 6.56e3·9^-s + 6.14e4i·11^-s + 2.82e4i·13^-s + 1.76e5·15^-s + 1.93e5·17^-s − 3.66e5i·19^-s − 6.20e5i·21^-s − 8.20e5·23^-s − 2.79e6·25^-s − 5.31e5i·27^-s − 4.44e6i·29^-s + 2.59e6·31^-s + ⋯

L(s) = 1

+ 0.577i·3^-s − 1.55i·5^-s − 1.20·7^-s − 0.333·9^-s + 1.26i·11^-s + 0.274i·13^-s + 0.899·15^-s + 0.560·17^-s − 0.645i·19^-s − 0.696i·21^-s − 0.611·23^-s − 1.42·25^-s − 0.192i·27^-s − 1.16i·29^-s + 0.505·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(10-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$384$ = $2^{7} \cdot 3$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$197.773$
Root analytic conductor:	$14.0632$
Motivic weight:	$9$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{384} (193, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 384,\ (\ :9/2),\ 0.707 + 0.707i)$

Particular Values

$L(5)$	$\approx$	$1.187929334$
$L(\frac12)$	$\approx$	$1.187929334$
$L(\frac{11}{2})$		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 - 81iT $
good	5	$ 1 + 2.17e3iT - 1.95e6T^{2} $
	7	$ 1 + 7.65e3T + 4.03e7T^{2} $
	11	$ 1 - 6.14e4iT - 2.35e9T^{2} $
	13	$ 1 - 2.82e4iT - 1.06e10T^{2} $
	17	$ 1 - 1.93e5T + 1.18e11T^{2} $
	19	$ 1 + 3.66e5iT - 3.22e11T^{2} $
	23	$ 1 + 8.20e5T + 1.80e12T^{2} $
	29	$ 1 + 4.44e6iT - 1.45e13T^{2} $
	31	$ 1 - 2.59e6T + 2.64e13T^{2} $
	37	$ 1 - 2.14e7iT - 1.29e14T^{2} $
	41	$ 1 + 3.15e7T + 3.27e14T^{2} $
	43	$ 1 - 9.42e6iT - 5.02e14T^{2} $
	47	$ 1 - 3.59e7T + 1.11e15T^{2} $
	53	$ 1 - 3.97e7iT - 3.29e15T^{2} $
	59	$ 1 - 9.36e7iT - 8.66e15T^{2} $
	61	$ 1 + 3.86e7iT - 1.16e16T^{2} $
	67	$ 1 - 1.80e8iT - 2.72e16T^{2} $
	71	$ 1 + 1.54e8T + 4.58e16T^{2} $
	73	$ 1 + 3.79e8T + 5.88e16T^{2} $
	79	$ 1 - 5.21e8T + 1.19e17T^{2} $
	83	$ 1 - 5.93e8iT - 1.86e17T^{2} $
	89	$ 1 - 1.05e8T + 3.50e17T^{2} $
	97	$ 1 - 2.91e8T + 7.60e17T^{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

−9.763525484431557966252207195347, −8.971873249842515947186874743946, −8.072777337170252694188348610498, −6.83944536301088795279046513581, −5.76485165642549027955528821978, −4.73621074316343202848814196691, −4.14751840736862415026289607540, −2.84590628906163553683020821598, −1.51013710833820625445684365554, −0.35436453757723016105357426356, 0.57948796004923128676837489042, 2.09683888611172886881250512250, 3.28024275217474971685397143220, 3.45575254095212519188365466745, 5.64335655488735262894080158284, 6.28248111373637640386380519568, 7.01256625033428807834185479635, 7.893459102711503117494186062005, 9.009163082580646331244030890310, 10.20052586425316762987944962314

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 $\F_{q}$

\(L(5)\)	\(\approx\)	\(1.187929334\)
\(L(\frac12)\)	\(\approx\)	\(1.187929334\)
\(L(\frac{11}{2})\)		not available
\(L(1)\)		not available

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