LMFDB - L-function 2-2e6-8.5-c21-0-30

Dirichlet series

L(s) = 1

− 5.84e4i·3^-s + 3.08e7i·5^-s − 2.12e7·7^-s + 7.04e9·9^-s − 3.98e10i·11^-s − 5.30e11i·13^-s + 1.80e12·15^-s + 8.07e12·17^-s + 3.47e12i·19^-s + 1.24e12i·21^-s + 1.48e14·23^-s − 4.77e14·25^-s − 1.02e15i·27^-s − 7.04e14i·29^-s − 2.40e15·31^-s + ⋯

L(s) = 1

− 0.571i·3^-s + 1.41i·5^-s − 0.0284·7^-s + 0.673·9^-s − 0.463i·11^-s − 1.06i·13^-s + 0.808·15^-s + 0.971·17^-s + 0.130i·19^-s + 0.0162i·21^-s + 0.745·23^-s − 1.00·25^-s − 0.956i·27^-s − 0.311i·29^-s − 0.526·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(22-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$64$ = $2^{6}$
Sign:	$0.258 + 0.965i$
Analytic conductor:	$178.865$
Root analytic conductor:	$13.3740$
Motivic weight:	$21$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{64} (33, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 64,\ (\ :21/2),\ 0.258 + 0.965i)$

Particular Values

$L(11)$	$\approx$	$2.056451104$
$L(\frac12)$	$\approx$	$2.056451104$
$L(\frac{23}{2})$		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 $
good	3	$ 1 + 5.84e4iT - 1.04e10T^{2} $
	5	$ 1 - 3.08e7iT - 4.76e14T^{2} $
	7	$ 1 + 2.12e7T + 5.58e17T^{2} $
	11	$ 1 + 3.98e10iT - 7.40e21T^{2} $
	13	$ 1 + 5.30e11iT - 2.47e23T^{2} $
	17	$ 1 - 8.07e12T + 6.90e25T^{2} $
	19	$ 1 - 3.47e12iT - 7.14e26T^{2} $
	23	$ 1 - 1.48e14T + 3.94e28T^{2} $
	29	$ 1 + 7.04e14iT - 5.13e30T^{2} $
	31	$ 1 + 2.40e15T + 2.08e31T^{2} $
	37	$ 1 - 3.01e16iT - 8.55e32T^{2} $
	41	$ 1 + 1.09e17T + 7.38e33T^{2} $
	43	$ 1 + 2.78e17iT - 2.00e34T^{2} $
	47	$ 1 + 2.62e17T + 1.30e35T^{2} $
	53	$ 1 + 2.30e17iT - 1.62e36T^{2} $
	59	$ 1 + 4.53e17iT - 1.54e37T^{2} $
	61	$ 1 - 5.41e18iT - 3.10e37T^{2} $
	67	$ 1 - 1.03e19iT - 2.22e38T^{2} $
	71	$ 1 - 4.98e19T + 7.52e38T^{2} $
	73	$ 1 - 2.15e19T + 1.34e39T^{2} $
	79	$ 1 - 5.01e18T + 7.08e39T^{2} $
	83	$ 1 + 2.43e20iT - 1.99e40T^{2} $
	89	$ 1 + 2.49e20T + 8.65e40T^{2} $
	97	$ 1 + 1.02e20T + 5.27e41T^{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

−10.59971671324216613467533648727, −9.934112781669979810588744865601, −8.218423198107134240342097627310, −7.28079213965380037707486375066, −6.51174032434352654551387394348, −5.33949921139876096959846000067, −3.59780994451895180455224035647, −2.85200605551163299322779007980, −1.59564801012529080856399764019, −0.42147691907448218555692595692, 1.00521764438416480189338982663, 1.78897953973850081312536500431, 3.56143240025617490250429645905, 4.60499858816428488975293634462, 5.18531616005637221059294737431, 6.76731662360893536383247590037, 8.030161205936989256139136758338, 9.242099571099261307804705485258, 9.723487657783364701289471859998, 11.14282189233460760340285534913

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 $\F_{q}$

\(L(11)\)	\(\approx\)	\(2.056451104\)
\(L(\frac12)\)	\(\approx\)	\(2.056451104\)
\(L(\frac{23}{2})\)		not available
\(L(1)\)		not available

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