LMFDB - L-function 2-2e5-4.3-c16-0-4

Dirichlet series

L(s) = 1

− 1.24e4i·3^-s − 5.59e5·5^-s + 8.71e6i·7^-s − 1.11e8·9^-s − 1.47e8i·11^-s + 4.14e8·13^-s + 6.95e9i·15^-s − 6.31e8·17^-s + 2.19e10i·19^-s + 1.08e11·21^-s − 7.67e10i·23^-s + 1.60e11·25^-s + 8.52e11i·27^-s + 1.23e11·29^-s − 1.90e11i·31^-s + ⋯

L(s) = 1

− 1.89i·3^-s − 1.43·5^-s + 1.51i·7^-s − 2.59·9^-s − 0.686i·11^-s + 0.508·13^-s + 2.71i·15^-s − 0.0905·17^-s + 1.29i·19^-s + 2.86·21^-s − 0.979i·23^-s + 1.04·25^-s + 3.01i·27^-s + 0.246·29^-s − 0.223i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(17-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$32$ = $2^{5}$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$51.9438$
Root analytic conductor:	$7.20720$
Motivic weight:	$16$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{32} (31, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 32,\ (\ :8),\ 0.707 + 0.707i)$

Particular Values

$L(\frac{17}{2})$	$\approx$	$1.007547162$
$L(\frac12)$	$\approx$	$1.007547162$
$L(9)$		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 $
good	3	$ 1 + 1.24e4iT - 4.30e7T^{2} $
	5	$ 1 + 5.59e5T + 1.52e11T^{2} $
	7	$ 1 - 8.71e6iT - 3.32e13T^{2} $
	11	$ 1 + 1.47e8iT - 4.59e16T^{2} $
	13	$ 1 - 4.14e8T + 6.65e17T^{2} $
	17	$ 1 + 6.31e8T + 4.86e19T^{2} $
	19	$ 1 - 2.19e10iT - 2.88e20T^{2} $
	23	$ 1 + 7.67e10iT - 6.13e21T^{2} $
	29	$ 1 - 1.23e11T + 2.50e23T^{2} $
	31	$ 1 + 1.90e11iT - 7.27e23T^{2} $
	37	$ 1 + 5.54e11T + 1.23e25T^{2} $
	41	$ 1 + 1.32e13T + 6.37e25T^{2} $
	43	$ 1 + 4.26e12iT - 1.36e26T^{2} $
	47	$ 1 - 8.72e12iT - 5.66e26T^{2} $
	53	$ 1 - 3.66e13T + 3.87e27T^{2} $
	59	$ 1 + 1.78e14iT - 2.15e28T^{2} $
	61	$ 1 - 2.89e14T + 3.67e28T^{2} $
	67	$ 1 - 1.70e14iT - 1.64e29T^{2} $
	71	$ 1 + 2.91e14iT - 4.16e29T^{2} $
	73	$ 1 - 1.24e15T + 6.50e29T^{2} $
	79	$ 1 - 1.72e15iT - 2.30e30T^{2} $
	83	$ 1 - 4.17e14iT - 5.07e30T^{2} $
	89	$ 1 + 3.35e15T + 1.54e31T^{2} $
	97	$ 1 + 6.85e15T + 6.14e31T^{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

−12.70800504334046819462381901778, −12.05486290711513256699518249201, −11.32313843439902066358274588193, −8.473802203284402731922901211792, −8.195591618294503718819373207463, −6.72862620168617901389096744341, −5.64531979636092292679654861362, −3.32533721233090034429773653616, −2.05747153682340354106658558534, −0.64302509466326376259400254133, 0.46502023751953663884970447002, 3.35869895965424607285416877588, 4.07752096843611863361843207020, 4.89089370096687217084201458506, 7.15338584363001012396915967469, 8.516347665602809376446659698438, 9.883588361475971481214946493752, 10.85961155799588195300359372728, 11.61962648183551165133214368404, 13.66426004629681264731906362692

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 $\F_{q}$

\(L(\frac{17}{2})\)	\(\approx\)	\(1.007547162\)
\(L(\frac12)\)	\(\approx\)	\(1.007547162\)
\(L(9)\)		not available
\(L(1)\)		not available

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