LMFDB - L-function 2-2900-5.4-c1-0-25

Dirichlet series

L(s) = 1

− 3.32i·3^-s + 1.32i·7^-s − 8.02·9^-s + 5.32·11^-s + 5.02i·13^-s − 6.34i·17^-s + 4.34·19^-s + 4.38·21^-s + 1.70i·23^-s + 16.6i·27^-s + 29^-s + 8.34·31^-s − 17.6i·33^-s − 6.93i·37^-s + 16.6·39^-s + ⋯

L(s) = 1

− 1.91i·3^-s + 0.499i·7^-s − 2.67·9^-s + 1.60·11^-s + 1.39i·13^-s − 1.53i·17^-s + 0.997·19^-s + 0.957·21^-s + 0.356i·23^-s + 3.21i·27^-s + 0.185·29^-s + 1.49·31^-s − 3.07i·33^-s − 1.14i·37^-s + 2.67·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2900$ = $2^{2} \cdot 5^{2} \cdot 29$
Sign:	$-0.447 + 0.894i$
Analytic conductor:	$23.1566$
Root analytic conductor:	$4.81213$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2900} (349, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2900,\ (\ :1/2),\ -0.447 + 0.894i)$

Particular Values

$L(1)$	$\approx$	$1.907991127$
$L(\frac12)$	$\approx$	$1.907991127$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
	29	$ 1 - T $
good	3	$ 1 + 3.32iT - 3T^{2} $
	7	$ 1 - 1.32iT - 7T^{2} $
	11	$ 1 - 5.32T + 11T^{2} $
	13	$ 1 - 5.02iT - 13T^{2} $
	17	$ 1 + 6.34iT - 17T^{2} $
	19	$ 1 - 4.34T + 19T^{2} $
	23	$ 1 - 1.70iT - 23T^{2} $
	31	$ 1 - 8.34T + 31T^{2} $
	37	$ 1 + 6.93iT - 37T^{2} $
	41	$ 1 + 1.02T + 41T^{2} $
	43	$ 1 + 10.7iT - 43T^{2} $
	47	$ 1 + 0.679iT - 47T^{2} $
	53	$ 1 + 2.38iT - 53T^{2} $
	59	$ 1 + 10.4T + 59T^{2} $
	61	$ 1 + 6.38T + 61T^{2} $
	67	$ 1 + 5.70iT - 67T^{2} $
	71	$ 1 + 3.61T + 71T^{2} $
	73	$ 1 - 6.73iT - 73T^{2} $
	79	$ 1 - 11.3T + 79T^{2} $
	83	$ 1 - 3.96iT - 83T^{2} $
	89	$ 1 + 2.58T + 89T^{2} $
	97	$ 1 + 15.3iT - 97T^{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

−8.560198263888960675763607158861, −7.42039572754321893419614562833, −7.10566472384534556754457525087, −6.42316289941697772913270207705, −5.78609873738579538250312131314, −4.75525054152828781931420743352, −3.47286901084952499372442918640, −2.46695073368184012527957424173, −1.66284136756615919445624543681, −0.74714789826024250908497805750, 1.08786333741248214862620771551, 2.97110453432116412779188712728, 3.51865179567782703564229555589, 4.31737817361596095833677590014, 4.85959900155327462509477696747, 5.95006218842075434436456532676, 6.35892965880746102328801111308, 7.81726887498035459006856975824, 8.419290309315023821305993803093, 9.168808684696348936758518713465

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.907991127\)
\(L(\frac12)\)	\(\approx\)	\(1.907991127\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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