LMFDB - L-function 2-29-29.28-c15-0-2

Dirichlet series

L(s) = 1

− 297. i·2^-s + 5.03e3i·3^-s − 5.56e4·4^-s − 1.22e5·5^-s + 1.49e6·6^-s + 2.97e6·7^-s + 6.80e6i·8^-s − 1.10e7·9^-s + 3.63e7i·10^-s + 6.17e6i·11^-s − 2.80e8i·12^-s − 4.98e7·13^-s − 8.83e8i·14^-s − 6.15e8i·15^-s + 2.00e8·16^-s − 2.30e9i·17^-s + ⋯

L(s) = 1

− 1.64i·2^-s + 1.32i·3^-s − 1.69·4^-s − 0.699·5^-s + 2.18·6^-s + 1.36·7^-s + 1.14i·8^-s − 0.766·9^-s + 1.14i·10^-s + 0.0954i·11^-s − 2.25i·12^-s − 0.220·13^-s − 2.23i·14^-s − 0.929i·15^-s + 0.186·16^-s − 1.36i·17^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(16-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$29$
Sign:	$0.315 - 0.948i$
Analytic conductor:	$41.3811$
Root analytic conductor:	$6.43281$
Motivic weight:	$15$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{29} (28, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 29,\ (\ :15/2),\ 0.315 - 0.948i)$

Particular Values

$L(8)$	$\approx$	$0.8139536054$
$L(\frac12)$	$\approx$	$0.8139536054$
$L(\frac{17}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	29	$ 1 + (-2.92e10 + 8.81e10i)T $
good	2	$ 1 + 297. iT - 3.27e4T^{2} $
	3	$ 1 - 5.03e3iT - 1.43e7T^{2} $
	5	$ 1 + 1.22e5T + 3.05e10T^{2} $
	7	$ 1 - 2.97e6T + 4.74e12T^{2} $
	11	$ 1 - 6.17e6iT - 4.17e15T^{2} $
	13	$ 1 + 4.98e7T + 5.11e16T^{2} $
	17	$ 1 + 2.30e9iT - 2.86e18T^{2} $
	19	$ 1 - 3.73e9iT - 1.51e19T^{2} $
	23	$ 1 + 2.07e9T + 2.66e20T^{2} $
	31	$ 1 - 1.24e11iT - 2.34e22T^{2} $
	37	$ 1 - 7.98e11iT - 3.33e23T^{2} $
	41	$ 1 - 7.89e11iT - 1.55e24T^{2} $
	43	$ 1 - 1.01e12iT - 3.17e24T^{2} $
	47	$ 1 - 1.01e12iT - 1.20e25T^{2} $
	53	$ 1 + 6.28e12T + 7.31e25T^{2} $
	59	$ 1 + 3.10e13T + 3.65e26T^{2} $
	61	$ 1 - 2.04e13iT - 6.02e26T^{2} $
	67	$ 1 - 1.18e12T + 2.46e27T^{2} $
	71	$ 1 + 1.12e14T + 5.87e27T^{2} $
	73	$ 1 + 5.76e13iT - 8.90e27T^{2} $
	79	$ 1 + 9.00e13iT - 2.91e28T^{2} $
	83	$ 1 + 1.65e14T + 6.11e28T^{2} $
	89	$ 1 - 8.14e14iT - 1.74e29T^{2} $
	97	$ 1 - 1.08e15iT - 6.33e29T^{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

−13.86238559612652539374077484623, −12.01644331180363423481093852855, −11.39336580587408539522433715327, −10.38026160511409456414017271304, −9.402782135607141364804709829038, −7.997170274379816679688465672781, −4.88961133383398656771255543503, −4.24547515302345766714022457138, −2.99168066581261082420451069178, −1.39438395201790389868887884930, 0.24018345214340489045207428010, 1.79507140251367049610924879796, 4.40054284971947537096268444120, 5.80873903871223830853841824081, 7.12181023462375346962017511358, 7.83413621174736903584028877488, 8.630848832116637286840585545090, 11.20666481387061300742663309129, 12.54100103886354981496505369295, 13.79861137472724909428414623660

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 $\F_{q}$

\(L(8)\)	\(\approx\)	\(0.8139536054\)
\(L(\frac12)\)	\(\approx\)	\(0.8139536054\)
\(L(\frac{17}{2})\)		not available
\(L(1)\)		not available

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