LMFDB - L-function 2-1900-95.94-c2-0-9

Dirichlet series

L(s) = 1

− 4.84·3^-s + 6.85i·7^-s + 14.4·9^-s − 9.59·11^-s + 10.0·13^-s + 27.8i·17^-s + (−17.5 + 7.24i)19^-s − 33.2i·21^-s + 8.14i·23^-s − 26.3·27^-s + 17.8i·29^-s − 20.3i·31^-s + 46.4·33^-s + 65.1·37^-s − 48.8·39^-s + ⋯

L(s) = 1

− 1.61·3^-s + 0.979i·7^-s + 1.60·9^-s − 0.872·11^-s + 0.775·13^-s + 1.63i·17^-s + (−0.924 + 0.381i)19^-s − 1.58i·21^-s + 0.354i·23^-s − 0.977·27^-s + 0.614i·29^-s − 0.657i·31^-s + 1.40·33^-s + 1.75·37^-s − 1.25·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0721i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1900$ = $2^{2} \cdot 5^{2} \cdot 19$
Sign:	$-0.997 - 0.0721i$
Analytic conductor:	$51.7712$
Root analytic conductor:	$7.19522$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1900} (949, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1900,\ (\ :1),\ -0.997 - 0.0721i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.5550704430$
$L(\frac12)$	$\approx$	$0.5550704430$
$L(2)$		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 $
	5	$ 1 $
	19	$ 1 + (17.5 - 7.24i)T $
good	3	$ 1 + 4.84T + 9T^{2} $
	7	$ 1 - 6.85iT - 49T^{2} $
	11	$ 1 + 9.59T + 121T^{2} $
	13	$ 1 - 10.0T + 169T^{2} $
	17	$ 1 - 27.8iT - 289T^{2} $
	23	$ 1 - 8.14iT - 529T^{2} $
	29	$ 1 - 17.8iT - 841T^{2} $
	31	$ 1 + 20.3iT - 961T^{2} $
	37	$ 1 - 65.1T + 1.36e3T^{2} $
	41	$ 1 - 16.0iT - 1.68e3T^{2} $
	43	$ 1 - 60.7iT - 1.84e3T^{2} $
	47	$ 1 + 9.60iT - 2.20e3T^{2} $
	53	$ 1 + 14.7T + 2.80e3T^{2} $
	59	$ 1 + 0.442iT - 3.48e3T^{2} $
	61	$ 1 - 65.2T + 3.72e3T^{2} $
	67	$ 1 - 103.T + 4.48e3T^{2} $
	71	$ 1 + 90.5iT - 5.04e3T^{2} $
	73	$ 1 - 88.8iT - 5.32e3T^{2} $
	79	$ 1 - 86.7iT - 6.24e3T^{2} $
	83	$ 1 + 47.4iT - 6.88e3T^{2} $
	89	$ 1 - 58.0iT - 7.92e3T^{2} $
	97	$ 1 + 165.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

−9.617687193510179852299212939096, −8.467990991051523580932127102995, −7.961240111339433118912036814479, −6.68165058204999425172413244182, −5.99278708851474142268318152901, −5.69153891070487991782303210523, −4.73291506466706827128027537609, −3.82370764598254430170687706684, −2.38066304959103132891630939769, −1.21010829102872371185098962751, 0.25151162305957827222130860700, 0.904554361178712409782859230467, 2.50853999555142953879743917789, 3.93032217189263753832602332475, 4.72231160381475017557216880255, 5.38257805239814665296969110979, 6.26442741821915306564573216794, 6.93832633449166062603393781316, 7.58996258187527034536152382779, 8.622236422577721480452354812192

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.5550704430\)
\(L(\frac12)\)	\(\approx\)	\(0.5550704430\)
\(L(2)\)		not available
\(L(1)\)		not available

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