LMFDB - L-function 2-384-8.5-c9-0-46

Dirichlet series

L(s) = 1

− 81i·3^-s − 553. i·5^-s + 1.24e4·7^-s − 6.56e3·9^-s + 3.97e4i·11^-s + 9.37e4i·13^-s − 4.48e4·15^-s − 2.11e5·17^-s − 2.01e5i·19^-s − 1.01e6i·21^-s − 1.90e6·23^-s + 1.64e6·25^-s + 5.31e5i·27^-s − 4.59e6i·29^-s + 1.17e6·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 0.396i·5^-s + 1.96·7^-s − 0.333·9^-s + 0.818i·11^-s + 0.910i·13^-s − 0.228·15^-s − 0.613·17^-s − 0.354i·19^-s − 1.13i·21^-s − 1.41·23^-s + 0.842·25^-s + 0.192i·27^-s − 1.20i·29^-s + 0.227·31^-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

$L(5)$	$\approx$	$2.950426562$
$L(\frac12)$	$\approx$	$2.950426562$
$L(\frac{11}{2})$		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 + 553. iT - 1.95e6T^{2} $
	7	$ 1 - 1.24e4T + 4.03e7T^{2} $
	11	$ 1 - 3.97e4iT - 2.35e9T^{2} $
	13	$ 1 - 9.37e4iT - 1.06e10T^{2} $
	17	$ 1 + 2.11e5T + 1.18e11T^{2} $
	19	$ 1 + 2.01e5iT - 3.22e11T^{2} $
	23	$ 1 + 1.90e6T + 1.80e12T^{2} $
	29	$ 1 + 4.59e6iT - 1.45e13T^{2} $
	31	$ 1 - 1.17e6T + 2.64e13T^{2} $
	37	$ 1 + 1.71e7iT - 1.29e14T^{2} $
	41	$ 1 - 7.69e6T + 3.27e14T^{2} $
	43	$ 1 - 3.79e7iT - 5.02e14T^{2} $
	47	$ 1 - 1.24e7T + 1.11e15T^{2} $
	53	$ 1 + 4.48e7iT - 3.29e15T^{2} $
	59	$ 1 + 4.27e7iT - 8.66e15T^{2} $
	61	$ 1 - 3.97e6iT - 1.16e16T^{2} $
	67	$ 1 + 7.11e6iT - 2.72e16T^{2} $
	71	$ 1 - 1.75e8T + 4.58e16T^{2} $
	73	$ 1 + 3.15e8T + 5.88e16T^{2} $
	79	$ 1 - 2.06e8T + 1.19e17T^{2} $
	83	$ 1 + 1.10e8iT - 1.86e17T^{2} $
	89	$ 1 + 7.60e8T + 3.50e17T^{2} $
	97	$ 1 - 1.20e9T + 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

−9.558547029985468986725702609063, −8.587724070065957181850515931070, −7.892994650760235525625545690066, −7.09429947132912327340714029686, −5.90756612331910111485792925577, −4.72025295271740783153991130035, −4.27183432480192449337044401890, −2.22580114212344125859384132009, −1.79053880031731331956095415423, −0.67920916625737333614039025373, 0.799924679040807008262850692497, 1.92671301421582370984473478778, 3.09019310334355565201452725120, 4.23948982121395960478471228462, 5.11602237382361518259048770062, 5.93066363109770926217710396581, 7.32883189621571338676398882814, 8.301213093253925247855232764696, 8.722611182074966840447738073557, 10.25293754873476181328994268852

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 $\F_{q}$

\(L(5)\)	\(\approx\)	\(2.950426562\)
\(L(\frac12)\)	\(\approx\)	\(2.950426562\)
\(L(\frac{11}{2})\)		not available
\(L(1)\)		not available

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