LMFDB - L-function 2-3200-40.29-c1-0-27

Dirichlet series

L(s) = 1

− 3·9^-s − 4·13^-s − 2i·17^-s + 4i·29^-s + 12·37^-s + 10·41^-s + 7·49^-s + 4·53^-s + 12i·61^-s − 6i·73^-s + 9·81^-s + 10·89^-s − 18i·97^-s + 20i·101^-s + 20i·109^-s + ⋯

L(s) = 1

− 9^-s − 1.10·13^-s − 0.485i·17^-s + 0.742i·29^-s + 1.97·37^-s + 1.56·41^-s + 49^-s + 0.549·53^-s + 1.53i·61^-s − 0.702i·73^-s + 81^-s + 1.05·89^-s − 1.82i·97^-s + 1.99i·101^-s + 1.91i·109^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3200$ = $2^{7} \cdot 5^{2}$
Sign:	$0.948 - 0.316i$
Analytic conductor:	$25.5521$
Root analytic conductor:	$5.05491$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3200} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3200,\ (\ :1/2),\ 0.948 - 0.316i)$

Particular Values

$L(1)$	$\approx$	$1.394487793$
$L(\frac12)$	$\approx$	$1.394487793$
$L(\frac{3}{2})$		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 $
good	3	$ 1 + 3T^{2} $
	7	$ 1 - 7T^{2} $
	11	$ 1 - 11T^{2} $
	13	$ 1 + 4T + 13T^{2} $
	17	$ 1 + 2iT - 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 - 23T^{2} $
	29	$ 1 - 4iT - 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 - 12T + 37T^{2} $
	41	$ 1 - 10T + 41T^{2} $
	43	$ 1 + 43T^{2} $
	47	$ 1 - 47T^{2} $
	53	$ 1 - 4T + 53T^{2} $
	59	$ 1 - 59T^{2} $
	61	$ 1 - 12iT - 61T^{2} $
	67	$ 1 + 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 6iT - 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 - 10T + 89T^{2} $
	97	$ 1 + 18iT - 97T^{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

−8.864229234684621713217378287232, −7.80852877735676114772836925711, −7.39303259468716917458983305383, −6.38193377788816080786625430517, −5.65653825307123333157078792124, −4.92256337003355570242357659489, −4.06320426163702879380594545379, −2.89680729236596222818665924399, −2.35382097286369136464435644832, −0.76679370213445048722903756590, 0.61710629357465666210310724734, 2.20606770134580424980998439924, 2.84565555392979675698912581105, 3.97484686586189709183783656791, 4.76958668077760820362136380922, 5.70510013203006760095319576936, 6.21341210768247059980174048170, 7.25438679845608531451070043837, 7.896149526479872609028048004255, 8.566839174917558726316243266225

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(1.394487793\)
\(L(\frac12)\)	\(\approx\)	\(1.394487793\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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