LMFDB - L-function 2-575-115.114-c0-0-6

Dirichlet series

L(s) = 1

− 0.347i·2^-s − 1.87i·3^-s + 0.879·4^-s − 0.652·6^-s − 0.652i·8^-s − 2.53·9^-s − 1.65i·12^-s + 1.53i·13^-s + 0.652·16^-s + 0.879i·18^-s + i·23^-s − 1.22·24^-s + 0.532·26^-s + 2.87i·27^-s − 0.347·29^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$575$ = $5^{2} \cdot 23$
Sign:	$-0.447 + 0.894i$
Analytic conductor:	$0.286962$
Root analytic conductor:	$0.535688$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{575} (574, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 575,\ (\ :0),\ -0.447 + 0.894i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.047700484$
$L(\frac12)$	$\approx$	$1.047700484$
$L(1)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	5	$ 1 $
	23	$ 1 - iT $
good	2	$ 1 + 0.347iT - T^{2} $
	3	$ 1 + 1.87iT - T^{2} $
	7	$ 1 + T^{2} $
	11	$ 1 - T^{2} $
	13	$ 1 - 1.53iT - T^{2} $
	17	$ 1 + T^{2} $
	19	$ 1 - T^{2} $
	29	$ 1 + 0.347T + T^{2} $
	31	$ 1 - 0.347T + T^{2} $
	37	$ 1 + T^{2} $
	41	$ 1 + 1.87T + T^{2} $
	43	$ 1 + T^{2} $
	47	$ 1 + 1.53iT - T^{2} $
	53	$ 1 + T^{2} $
	59	$ 1 - T + T^{2} $
	61	$ 1 - T^{2} $
	67	$ 1 + T^{2} $
	71	$ 1 - 1.53T + T^{2} $
	73	$ 1 - 0.347iT - T^{2} $
	79	$ 1 - T^{2} $
	83	$ 1 + T^{2} $
	89	$ 1 - T^{2} $
	97	$ 1 + T^{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

−11.14624939606953191562356723784, −9.818731931114226062446276030208, −8.671416731878316399030280621035, −7.77213228551006311876339338253, −6.86296820134107009391661904743, −6.59249482239255692837004953638, −5.43481611544346874040195694756, −3.51182357714868104001628295666, −2.23332413431380144394148734793, −1.51211610996373715401798546901, 2.70668402787988307654011355282, 3.53603351599010303788478917421, 4.84428056529770984461434374852, 5.55682429534839748070404412341, 6.48749786562062392252703642416, 7.929838568345376469181868572800, 8.551588867909273083740978035698, 9.720326212894142164461926211291, 10.39201381465485237793770973300, 10.92403484226613396039832546746

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(1.047700484\)
\(L(\frac12)\)	\(\approx\)	\(1.047700484\)
\(L(1)\)		not available
\(L(1)\)		not available

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