LMFDB - L-function 2-384-8.5-c9-0-53

Dirichlet series

L(s) = 1

− 81i·3^-s + 1.87e3i·5^-s + 4.37e3·7^-s − 6.56e3·9^-s + 1.08e4i·11^-s − 1.35e5i·13^-s + 1.52e5·15^-s + 5.48e5·17^-s − 9.50e3i·19^-s − 3.54e5i·21^-s + 2.26e6·23^-s − 1.57e6·25^-s + 5.31e5i·27^-s + 5.80e5i·29^-s − 9.70e6·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 1.34i·5^-s + 0.689·7^-s − 0.333·9^-s + 0.223i·11^-s − 1.31i·13^-s + 0.775·15^-s + 1.59·17^-s − 0.0167i·19^-s − 0.397i·21^-s + 1.68·23^-s − 0.806·25^-s + 0.192i·27^-s + 0.152i·29^-s − 1.88·31^-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

$L(5)$	$\approx$	$2.546303373$
$L(\frac12)$	$\approx$	$2.546303373$
$L(\frac{11}{2})$		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 - 1.87e3iT - 1.95e6T^{2} $
	7	$ 1 - 4.37e3T + 4.03e7T^{2} $
	11	$ 1 - 1.08e4iT - 2.35e9T^{2} $
	13	$ 1 + 1.35e5iT - 1.06e10T^{2} $
	17	$ 1 - 5.48e5T + 1.18e11T^{2} $
	19	$ 1 + 9.50e3iT - 3.22e11T^{2} $
	23	$ 1 - 2.26e6T + 1.80e12T^{2} $
	29	$ 1 - 5.80e5iT - 1.45e13T^{2} $
	31	$ 1 + 9.70e6T + 2.64e13T^{2} $
	37	$ 1 + 6.94e6iT - 1.29e14T^{2} $
	41	$ 1 - 2.68e7T + 3.27e14T^{2} $
	43	$ 1 + 3.81e7iT - 5.02e14T^{2} $
	47	$ 1 + 2.91e7T + 1.11e15T^{2} $
	53	$ 1 - 2.73e7iT - 3.29e15T^{2} $
	59	$ 1 + 1.79e8iT - 8.66e15T^{2} $
	61	$ 1 - 7.24e7iT - 1.16e16T^{2} $
	67	$ 1 - 1.17e8iT - 2.72e16T^{2} $
	71	$ 1 + 1.65e8T + 4.58e16T^{2} $
	73	$ 1 + 1.98e8T + 5.88e16T^{2} $
	79	$ 1 + 4.79e7T + 1.19e17T^{2} $
	83	$ 1 + 1.91e8iT - 1.86e17T^{2} $
	89	$ 1 + 3.89e8T + 3.50e17T^{2} $
	97	$ 1 - 1.96e8T + 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

−9.878628097384694232125822341832, −8.622090935122760963691323328480, −7.39744361831374778171726728220, −7.34561107252451245855275393676, −5.91820413495553641769661564615, −5.18505576259490294598848655843, −3.49377505286636225972765729332, −2.83619826414574872292649850558, −1.66038246586228707360176385212, −0.54210359830131275106757978877, 0.927796451695354100505256672215, 1.63301132454795086185474937487, 3.21144697028291759093652940823, 4.39701774607293506553436350618, 4.98644428394005553645656621345, 5.85946666980019895416083727201, 7.32660609531659026003059497083, 8.294155431108621952721478083601, 9.109583863136514874007306082057, 9.632598376265279576167117754006

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 $\F_{q}$

\(L(5)\)	\(\approx\)	\(2.546303373\)
\(L(\frac12)\)	\(\approx\)	\(2.546303373\)
\(L(\frac{11}{2})\)		not available
\(L(1)\)		not available

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