LMFDB - L-function 2-920-23.22-c2-0-44

Dirichlet series

L(s) = 1

− 1.88·3^-s − 2.23i·5^-s − 13.3i·7^-s − 5.45·9^-s − 18.2i·11^-s + 7.91·13^-s + 4.20i·15^-s − 16.9i·17^-s + 19.4i·19^-s + 25.2i·21^-s + (4.24 − 22.6i)23^-s − 5.00·25^-s + 27.2·27^-s − 41.4·29^-s + 3.51·31^-s + ⋯

L(s) = 1

− 0.627·3^-s − 0.447i·5^-s − 1.91i·7^-s − 0.606·9^-s − 1.65i·11^-s + 0.609·13^-s + 0.280i·15^-s − 0.995i·17^-s + 1.02i·19^-s + 1.20i·21^-s + (0.184 − 0.982i)23^-s − 0.200·25^-s + 1.00·27^-s − 1.42·29^-s + 0.113·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.184i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$920$ = $2^{3} \cdot 5 \cdot 23$
Sign:	$-0.982 - 0.184i$
Analytic conductor:	$25.0681$
Root analytic conductor:	$5.00681$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{920} (321, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 920,\ (\ :1),\ -0.982 - 0.184i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.8834143078$
$L(\frac12)$	$\approx$	$0.8834143078$
$L(2)$		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 $
	5	$ 1 + 2.23iT $
	23	$ 1 + (-4.24 + 22.6i)T $
good	3	$ 1 + 1.88T + 9T^{2} $
	7	$ 1 + 13.3iT - 49T^{2} $
	11	$ 1 + 18.2iT - 121T^{2} $
	13	$ 1 - 7.91T + 169T^{2} $
	17	$ 1 + 16.9iT - 289T^{2} $
	19	$ 1 - 19.4iT - 361T^{2} $
	29	$ 1 + 41.4T + 841T^{2} $
	31	$ 1 - 3.51T + 961T^{2} $
	37	$ 1 - 63.0iT - 1.36e3T^{2} $
	41	$ 1 - 54.7T + 1.68e3T^{2} $
	43	$ 1 - 26.6iT - 1.84e3T^{2} $
	47	$ 1 - 7.59T + 2.20e3T^{2} $
	53	$ 1 + 37.5iT - 2.80e3T^{2} $
	59	$ 1 - 56.8T + 3.48e3T^{2} $
	61	$ 1 + 110. iT - 3.72e3T^{2} $
	67	$ 1 + 22.9iT - 4.48e3T^{2} $
	71	$ 1 - 1.04T + 5.04e3T^{2} $
	73	$ 1 + 53.8T + 5.32e3T^{2} $
	79	$ 1 - 30.8iT - 6.24e3T^{2} $
	83	$ 1 - 43.6iT - 6.88e3T^{2} $
	89	$ 1 + 17.8iT - 7.92e3T^{2} $
	97	$ 1 - 61.4iT - 9.40e3T^{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

−9.543930374524662323573949738866, −8.434240880214206273464458333985, −7.87080046156648360126413943439, −6.72119096565735626234010835334, −6.00728318690096297468297016533, −5.06356176237061172021529657296, −4.00573539985637602660342297824, −3.16569432813760095718772786864, −1.08957216435886790148542455082, −0.35779366855508506807192804025, 1.90787689813530107626896229295, 2.73993314273516338779054785361, 4.15650324530712442101967803239, 5.51335780068115969755785921940, 5.71878162041847680572643808249, 6.80658805925257985826521402189, 7.74430142902088576086778490436, 8.967960049919786662303698555282, 9.240523368053489698704615886542, 10.45117705613624385575903836335

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.8834143078\)
\(L(\frac12)\)	\(\approx\)	\(0.8834143078\)
\(L(2)\)		not available
\(L(1)\)		not available

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