LMFDB - L-function 2-384-8.5-c9-0-25

Dirichlet series

L(s) = 1

− 81i·3^-s + 473. i·5^-s − 574.·7^-s − 6.56e3·9^-s + 7.27e4i·11^-s − 1.48e4i·13^-s + 3.83e4·15^-s + 1.09e5·17^-s − 4.97e5i·19^-s + 4.65e4i·21^-s − 1.04e5·23^-s + 1.72e6·25^-s + 5.31e5i·27^-s − 7.06e6i·29^-s − 2.30e6·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.338i·5^-s − 0.0905·7^-s − 0.333·9^-s + 1.49i·11^-s − 0.144i·13^-s + 0.195·15^-s + 0.319·17^-s − 0.875i·19^-s + 0.0522i·21^-s − 0.0780·23^-s + 0.885·25^-s + 0.192i·27^-s − 1.85i·29^-s − 0.448·31^-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

$L(5)$	$\approx$	$1.782778153$
$L(\frac12)$	$\approx$	$1.782778153$
$L(\frac{11}{2})$		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 - 473. iT - 1.95e6T^{2} $
	7	$ 1 + 574.T + 4.03e7T^{2} $
	11	$ 1 - 7.27e4iT - 2.35e9T^{2} $
	13	$ 1 + 1.48e4iT - 1.06e10T^{2} $
	17	$ 1 - 1.09e5T + 1.18e11T^{2} $
	19	$ 1 + 4.97e5iT - 3.22e11T^{2} $
	23	$ 1 + 1.04e5T + 1.80e12T^{2} $
	29	$ 1 + 7.06e6iT - 1.45e13T^{2} $
	31	$ 1 + 2.30e6T + 2.64e13T^{2} $
	37	$ 1 - 5.20e6iT - 1.29e14T^{2} $
	41	$ 1 + 1.08e7T + 3.27e14T^{2} $
	43	$ 1 + 2.04e7iT - 5.02e14T^{2} $
	47	$ 1 - 3.47e7T + 1.11e15T^{2} $
	53	$ 1 - 9.34e7iT - 3.29e15T^{2} $
	59	$ 1 - 9.04e7iT - 8.66e15T^{2} $
	61	$ 1 + 2.00e7iT - 1.16e16T^{2} $
	67	$ 1 + 1.09e8iT - 2.72e16T^{2} $
	71	$ 1 - 2.65e8T + 4.58e16T^{2} $
	73	$ 1 - 5.05e7T + 5.88e16T^{2} $
	79	$ 1 + 1.24e7T + 1.19e17T^{2} $
	83	$ 1 - 2.72e8iT - 1.86e17T^{2} $
	89	$ 1 + 2.70e8T + 3.50e17T^{2} $
	97	$ 1 + 4.06e8T + 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

−9.923388870652858383313233514075, −9.060988866793225241945901202027, −7.87644788940118557767895079468, −7.15376756025456075112219247016, −6.38756317689595828840468083374, −5.18925180660696784383106623452, −4.16545162140183838018348106344, −2.80930781000444961886047465498, −1.99251389552192638117603098122, −0.78464321964758404699970085552, 0.41667536918581748038528312373, 1.50891838598974429958691008274, 3.05666383082744309199179626362, 3.73073818384847725015953873396, 5.01170105915671368647078147615, 5.74628288899127445360026059299, 6.81244433751092106398524884663, 8.154585389704249016288044123102, 8.756092933277069396575757709207, 9.666119489714531236471145295099

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 $\F_{q}$

\(L(5)\)	\(\approx\)	\(1.782778153\)
\(L(\frac12)\)	\(\approx\)	\(1.782778153\)
\(L(\frac{11}{2})\)		not available
\(L(1)\)		not available

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