LMFDB - L-function 2-1900-95.94-c2-0-20

Dirichlet series

L(s) = 1

− 3.53·3^-s − 0.468i·7^-s + 3.51·9^-s − 14.1·11^-s − 11.0·13^-s + 22.0i·17^-s + (−16.8 + 8.79i)19^-s + 1.65i·21^-s − 41.1i·23^-s + 19.4·27^-s + 46.7i·29^-s + 33.9i·31^-s + 49.9·33^-s − 62.6·37^-s + 39.2·39^-s + ⋯

L(s) = 1

− 1.17·3^-s − 0.0669i·7^-s + 0.390·9^-s − 1.28·11^-s − 0.853·13^-s + 1.29i·17^-s + (−0.886 + 0.462i)19^-s + 0.0789i·21^-s − 1.79i·23^-s + 0.718·27^-s + 1.61i·29^-s + 1.09i·31^-s + 1.51·33^-s − 1.69·37^-s + 1.00·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1900$ = $2^{2} \cdot 5^{2} \cdot 19$
Sign:	$0.810 + 0.585i$
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:	$(2,\ 1900,\ (\ :1),\ 0.810 + 0.585i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.4051548786$
$L(\frac12)$	$\approx$	$0.4051548786$
$L(2)$		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 + (16.8 - 8.79i)T $
good	3	$ 1 + 3.53T + 9T^{2} $
	7	$ 1 + 0.468iT - 49T^{2} $
	11	$ 1 + 14.1T + 121T^{2} $
	13	$ 1 + 11.0T + 169T^{2} $
	17	$ 1 - 22.0iT - 289T^{2} $
	23	$ 1 + 41.1iT - 529T^{2} $
	29	$ 1 - 46.7iT - 841T^{2} $
	31	$ 1 - 33.9iT - 961T^{2} $
	37	$ 1 + 62.6T + 1.36e3T^{2} $
	41	$ 1 + 23.5iT - 1.68e3T^{2} $
	43	$ 1 + 32.8iT - 1.84e3T^{2} $
	47	$ 1 + 43.2iT - 2.20e3T^{2} $
	53	$ 1 + 99.9T + 2.80e3T^{2} $
	59	$ 1 - 66.7iT - 3.48e3T^{2} $
	61	$ 1 + 36.0T + 3.72e3T^{2} $
	67	$ 1 - 93.5T + 4.48e3T^{2} $
	71	$ 1 - 124. iT - 5.04e3T^{2} $
	73	$ 1 - 41.3iT - 5.32e3T^{2} $
	79	$ 1 + 27.2iT - 6.24e3T^{2} $
	83	$ 1 + 32.9iT - 6.88e3T^{2} $
	89	$ 1 + 86.3iT - 7.92e3T^{2} $
	97	$ 1 - 20.8T + 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

−8.691724736494264221077407634038, −8.347953952977635381052506059450, −7.10962410958093486578545551864, −6.59986362345028106859073250232, −5.57536897975525090647669845723, −5.12625350297352049263713155501, −4.22048751833256471734107127905, −2.94678015644865472059892862219, −1.81172493912849204692220761992, −0.25615099291089539653920209865, 0.47214322852351777726885865627, 2.14559598616070637185880203637, 3.09530099320231704260769449912, 4.55960273100382460836707861591, 5.11897039731082115333271458283, 5.79074921347153541576064244336, 6.63522791651380803387140488605, 7.54439398313744233299295842067, 8.079923365135098157509032872863, 9.441260659082869154449552609866

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.4051548786\)
\(L(\frac12)\)	\(\approx\)	\(0.4051548786\)
\(L(2)\)		not available
\(L(1)\)		not available

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