LMFDB - L-function 2-920-5.4-c1-0-15

Dirichlet series

L(s) = 1

− 2i·3^-s + (2 + i)5^-s + 3i·7^-s − 9^-s − 6i·13^-s + (2 − 4i)15^-s + 7i·17^-s + 4·19^-s + 6·21^-s − i·23^-s + (3 + 4i)25^-s − 4i·27^-s + 9·29^-s − 3·31^-s + (−3 + 6i)35^-s + ⋯

L(s) = 1

− 1.15i·3^-s + (0.894 + 0.447i)5^-s + 1.13i·7^-s − 0.333·9^-s − 1.66i·13^-s + (0.516 − 1.03i)15^-s + 1.69i·17^-s + 0.917·19^-s + 1.30·21^-s − 0.208i·23^-s + (0.600 + 0.800i)25^-s − 0.769i·27^-s + 1.67·29^-s − 0.538·31^-s + (−0.507 + 1.01i)35^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$920$ = $2^{3} \cdot 5 \cdot 23$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$7.34623$
Root analytic conductor:	$2.71039$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{920} (369, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 920,\ (\ :1/2),\ 0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$1.90184 - 0.448965i$
$L(\frac12)$	$\approx$	$1.90184 - 0.448965i$
$L(\frac{3}{2})$		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 + (-2 - i)T $
	23	$ 1 + iT $
good	3	$ 1 + 2iT - 3T^{2} $
	7	$ 1 - 3iT - 7T^{2} $
	11	$ 1 + 11T^{2} $
	13	$ 1 + 6iT - 13T^{2} $
	17	$ 1 - 7iT - 17T^{2} $
	19	$ 1 - 4T + 19T^{2} $
	29	$ 1 - 9T + 29T^{2} $
	31	$ 1 + 3T + 31T^{2} $
	37	$ 1 - 7iT - 37T^{2} $
	41	$ 1 - 9T + 41T^{2} $
	43	$ 1 + 4iT - 43T^{2} $
	47	$ 1 - 2iT - 47T^{2} $
	53	$ 1 + 7iT - 53T^{2} $
	59	$ 1 + 9T + 59T^{2} $
	61	$ 1 + 2T + 61T^{2} $
	67	$ 1 + 13iT - 67T^{2} $
	71	$ 1 + 13T + 71T^{2} $
	73	$ 1 + 4iT - 73T^{2} $
	79	$ 1 - 2T + 79T^{2} $
	83	$ 1 + 11iT - 83T^{2} $
	89	$ 1 - 10T + 89T^{2} $
	97	$ 1 + 2iT - 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

−10.16740567927594096190764852702, −9.078263576763576428455897395614, −8.196658317727756488962915606710, −7.56119391130793280560037088721, −6.30656364495163437392375721977, −6.03808524500804080236427592142, −5.06330987539490000046857100460, −3.21039594208399749656021357050, −2.36153445555691264568359490626, −1.27796063889217452104315042610, 1.19881395260270307002118936162, 2.79929210880775103041963617438, 4.18706038023156161642051343076, 4.61023167475323989110401084390, 5.56877495154739735307388811593, 6.80303444331832880486229207689, 7.45642627460584773199300115106, 8.976314130652918751785312420792, 9.400650950863595796719640671996, 9.970993032740773998571020453231

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.90184 - 0.448965i\)
\(L(\frac12)\)	\(\approx\)	\(1.90184 - 0.448965i\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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