LMFDB - L-function 2-1900-19.18-c2-0-39

Dirichlet series

L(s) = 1

− 1.84i·3^-s − 10.5·7^-s + 5.60·9^-s + 12.7·11^-s − 23.0i·13^-s + 4.77·17^-s + (1.66 + 18.9i)19^-s + 19.4i·21^-s + 35.7·23^-s − 26.9i·27^-s + 13.7i·29^-s + 30.5i·31^-s − 23.4i·33^-s + 54.8i·37^-s − 42.4·39^-s + ⋯

L(s) = 1

− 0.614i·3^-s − 1.50·7^-s + 0.622·9^-s + 1.15·11^-s − 1.77i·13^-s + 0.280·17^-s + (0.0878 + 0.996i)19^-s + 0.924i·21^-s + 1.55·23^-s − 0.996i·27^-s + 0.474i·29^-s + 0.984i·31^-s − 0.710i·33^-s + 1.48i·37^-s − 1.08·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0878 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0878 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1900$ = $2^{2} \cdot 5^{2} \cdot 19$
Sign:	$0.0878 + 0.996i$
Analytic conductor:	$51.7712$
Root analytic conductor:	$7.19522$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1900} (1101, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1900,\ (\ :1),\ 0.0878 + 0.996i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.851064706$
$L(\frac12)$	$\approx$	$1.851064706$
$L(2)$		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 + (-1.66 - 18.9i)T $
good	3	$ 1 + 1.84iT - 9T^{2} $
	7	$ 1 + 10.5T + 49T^{2} $
	11	$ 1 - 12.7T + 121T^{2} $
	13	$ 1 + 23.0iT - 169T^{2} $
	17	$ 1 - 4.77T + 289T^{2} $
	23	$ 1 - 35.7T + 529T^{2} $
	29	$ 1 - 13.7iT - 841T^{2} $
	31	$ 1 - 30.5iT - 961T^{2} $
	37	$ 1 - 54.8iT - 1.36e3T^{2} $
	41	$ 1 + 18.6iT - 1.68e3T^{2} $
	43	$ 1 + 29.0T + 1.84e3T^{2} $
	47	$ 1 + 11.3T + 2.20e3T^{2} $
	53	$ 1 + 10.7iT - 2.80e3T^{2} $
	59	$ 1 + 54.5iT - 3.48e3T^{2} $
	61	$ 1 - 17.7T + 3.72e3T^{2} $
	67	$ 1 + 60.2iT - 4.48e3T^{2} $
	71	$ 1 + 39.9iT - 5.04e3T^{2} $
	73	$ 1 - 24.6T + 5.32e3T^{2} $
	79	$ 1 - 6.95iT - 6.24e3T^{2} $
	83	$ 1 - 130.T + 6.88e3T^{2} $
	89	$ 1 + 160. iT - 7.92e3T^{2} $
	97	$ 1 + 71.2iT - 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

−8.830272446468344658435057011687, −8.008086354471286225818011231805, −7.07783121177345307259309042841, −6.60696002310476738354547443233, −5.85638040453553565863360352369, −4.83508027776315439614885142288, −3.42194197335934053000188274592, −3.17983003472400384995484689438, −1.53855369864986395055114349013, −0.60253667007943203612702077382, 1.00033938137239520477634462319, 2.40291184679900044452807678017, 3.61341883858328625123039072216, 4.09998230629474758801194129632, 5.01077572656205199441657027099, 6.29019141855709737670166894104, 6.76313135067287445926551177592, 7.35823640603198821296142600690, 8.966899211997750114449974190370, 9.276314587982048082557992587242

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.851064706\)
\(L(\frac12)\)	\(\approx\)	\(1.851064706\)
\(L(2)\)		not available
\(L(1)\)		not available

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