LMFDB - L-function 2-320-40.29-c5-0-56

Dirichlet series

L(s) = 1

− 55.9·5^-s − 245. i·7^-s − 243·9^-s − 802i·11^-s + 245.·13^-s − 1.91e3i·19^-s + 3.33e3i·23^-s + 3.12e3·25^-s + 1.37e4i·35^-s − 1.58e4·37^-s + 1.52e4·41^-s + 1.35e4·45^-s − 1.53e4i·47^-s − 4.36e4·49^-s − 1.68e4·53^-s + ⋯

L(s) = 1

− 0.999·5^-s − 1.89i·7^-s − 9^-s − 1.99i·11^-s + 0.403·13^-s − 1.21i·19^-s + 1.31i·23^-s + 25^-s + 1.89i·35^-s − 1.90·37^-s + 1.41·41^-s + 0.999·45^-s − 1.01i·47^-s − 2.59·49^-s − 0.823·53^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$320$ = $2^{6} \cdot 5$
Sign:	$-0.707 - 0.707i$
Analytic conductor:	$51.3228$
Root analytic conductor:	$7.16399$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{320} (289, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 320,\ (\ :5/2),\ -0.707 - 0.707i)$

Particular Values

$L(3)$	$\approx$	$0.6121463614$
$L(\frac12)$	$\approx$	$0.6121463614$
$L(\frac{7}{2})$		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 + 55.9T $
good	3	$ 1 + 243T^{2} $
	7	$ 1 + 245. iT - 1.68e4T^{2} $
	11	$ 1 + 802iT - 1.61e5T^{2} $
	13	$ 1 - 245.T + 3.71e5T^{2} $
	17	$ 1 - 1.41e6T^{2} $
	19	$ 1 + 1.91e3iT - 2.47e6T^{2} $
	23	$ 1 - 3.33e3iT - 6.43e6T^{2} $
	29	$ 1 - 2.05e7T^{2} $
	31	$ 1 + 2.86e7T^{2} $
	37	$ 1 + 1.58e4T + 6.93e7T^{2} $
	41	$ 1 - 1.52e4T + 1.15e8T^{2} $
	43	$ 1 + 1.47e8T^{2} $
	47	$ 1 + 1.53e4iT - 2.29e8T^{2} $
	53	$ 1 + 1.68e4T + 4.18e8T^{2} $
	59	$ 1 + 2.79e4iT - 7.14e8T^{2} $
	61	$ 1 - 8.44e8T^{2} $
	67	$ 1 + 1.35e9T^{2} $
	71	$ 1 + 1.80e9T^{2} $
	73	$ 1 - 2.07e9T^{2} $
	79	$ 1 + 3.07e9T^{2} $
	83	$ 1 + 3.93e9T^{2} $
	89	$ 1 - 1.28e5T + 5.58e9T^{2} $
	97	$ 1 - 8.58e9T^{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

−10.65152090296737630634840641836, −9.071612021358279652302542260646, −8.235202846878292366075490308459, −7.42923494378033084855033595962, −6.41801082014443549506739662067, −5.11322377232845712621327229034, −3.72183536243922442378323533308, −3.27258461960101866368017691321, −0.883094249850789211345297601727, −0.20895829852626990355225425014, 1.94739011034032066857403606604, 2.97390815804244769131848872014, 4.40927904992655342356013119282, 5.44310181677161799662068178105, 6.48234610938060760380199711343, 7.76183234829800881270918482446, 8.570027047550695215927031553190, 9.292820825268561576898158261374, 10.52787881901438152501765879441, 11.63257071448113926146192372711

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(0.6121463614\)
\(L(\frac12)\)	\(\approx\)	\(0.6121463614\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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