LMFDB - L-function 2-1568-8.3-c2-0-24

Dirichlet series

L(s) = 1

− 4.50·3^-s − 7.01i·5^-s + 11.2·9^-s − 15.8·11^-s − 2.90i·13^-s + 31.5i·15^-s + 3.31·17^-s + 16.2·19^-s + 19.0i·23^-s − 24.1·25^-s − 10.3·27^-s + 21.1i·29^-s + 27.3i·31^-s + 71.2·33^-s + 17.6i·37^-s + ⋯

L(s) = 1

− 1.50·3^-s − 1.40i·5^-s + 1.25·9^-s − 1.43·11^-s − 0.223i·13^-s + 2.10i·15^-s + 0.194·17^-s + 0.853·19^-s + 0.827i·23^-s − 0.965·25^-s − 0.383·27^-s + 0.728i·29^-s + 0.882i·31^-s + 2.15·33^-s + 0.477i·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1568$ = $2^{5} \cdot 7^{2}$
Sign:	$0.876 + 0.481i$
Analytic conductor:	$42.7249$
Root analytic conductor:	$6.53642$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1568} (687, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1568,\ (\ :1),\ 0.876 + 0.481i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.7491500995$
$L(\frac12)$	$\approx$	$0.7491500995$
$L(2)$		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 $
	7	$ 1 $
good	3	$ 1 + 4.50T + 9T^{2} $
	5	$ 1 + 7.01iT - 25T^{2} $
	11	$ 1 + 15.8T + 121T^{2} $
	13	$ 1 + 2.90iT - 169T^{2} $
	17	$ 1 - 3.31T + 289T^{2} $
	19	$ 1 - 16.2T + 361T^{2} $
	23	$ 1 - 19.0iT - 529T^{2} $
	29	$ 1 - 21.1iT - 841T^{2} $
	31	$ 1 - 27.3iT - 961T^{2} $
	37	$ 1 - 17.6iT - 1.36e3T^{2} $
	41	$ 1 + 1.13T + 1.68e3T^{2} $
	43	$ 1 + 50.2T + 1.84e3T^{2} $
	47	$ 1 - 0.759iT - 2.20e3T^{2} $
	53	$ 1 - 44.9iT - 2.80e3T^{2} $
	59	$ 1 - 2.38T + 3.48e3T^{2} $
	61	$ 1 + 99.4iT - 3.72e3T^{2} $
	67	$ 1 - 66.4T + 4.48e3T^{2} $
	71	$ 1 - 44.4iT - 5.04e3T^{2} $
	73	$ 1 + 1.71T + 5.32e3T^{2} $
	79	$ 1 + 72.6iT - 6.24e3T^{2} $
	83	$ 1 + 102.T + 6.88e3T^{2} $
	89	$ 1 - 61.3T + 7.92e3T^{2} $
	97	$ 1 - 102.T + 9.40e3T^{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

−9.269786026271407894329349243414, −8.315020848060760680004234034766, −7.60527406901386138464211675760, −6.61445492208886607099179193251, −5.43798419520341047451712505378, −5.30422446059219608084828839782, −4.61975541612961219264132178128, −3.21533951899199645223851324373, −1.52344942439515823925878399981, −0.56853040022988610654684602382, 0.49218367324986981730460974828, 2.25607992583420958966502670226, 3.20860597368694672412001423279, 4.47555042183049137233869813959, 5.41345217425743131544312194517, 5.99658252793289766458469908144, 6.83098038538419951935065093316, 7.40864931905502387775524907122, 8.306202732423056860191109316746, 9.809300654900015342720160241998

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.7491500995\)
\(L(\frac12)\)	\(\approx\)	\(0.7491500995\)
\(L(2)\)		not available
\(L(1)\)		not available

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