LMFDB - L-function 2-1900-95.94-c2-0-23

Dirichlet series

L(s) = 1

− 2.93·3^-s − 5.62i·7^-s − 0.402·9^-s − 17.4·11^-s − 7.35·13^-s + 25.3i·17^-s + (2.59 + 18.8i)19^-s + 16.4i·21^-s + 19.3i·23^-s + 27.5·27^-s + 4.89i·29^-s − 45.2i·31^-s + 51.1·33^-s − 25.2·37^-s + 21.5·39^-s + ⋯

L(s) = 1

− 0.977·3^-s − 0.803i·7^-s − 0.0446·9^-s − 1.58·11^-s − 0.565·13^-s + 1.48i·17^-s + (0.136 + 0.990i)19^-s + 0.785i·21^-s + 0.839i·23^-s + 1.02·27^-s + 0.168i·29^-s − 1.45i·31^-s + 1.55·33^-s − 0.683·37^-s + 0.553·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1900$ = $2^{2} \cdot 5^{2} \cdot 19$
Sign:	$0.565 + 0.824i$
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:	$(2,\ 1900,\ (\ :1),\ 0.565 + 0.824i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.5487537505$
$L(\frac12)$	$\approx$	$0.5487537505$
$L(2)$		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 + (-2.59 - 18.8i)T $
good	3	$ 1 + 2.93T + 9T^{2} $
	7	$ 1 + 5.62iT - 49T^{2} $
	11	$ 1 + 17.4T + 121T^{2} $
	13	$ 1 + 7.35T + 169T^{2} $
	17	$ 1 - 25.3iT - 289T^{2} $
	23	$ 1 - 19.3iT - 529T^{2} $
	29	$ 1 - 4.89iT - 841T^{2} $
	31	$ 1 + 45.2iT - 961T^{2} $
	37	$ 1 + 25.2T + 1.36e3T^{2} $
	41	$ 1 - 16.2iT - 1.68e3T^{2} $
	43	$ 1 - 28.0iT - 1.84e3T^{2} $
	47	$ 1 - 38.9iT - 2.20e3T^{2} $
	53	$ 1 - 81.9T + 2.80e3T^{2} $
	59	$ 1 + 61.0iT - 3.48e3T^{2} $
	61	$ 1 + 94.1T + 3.72e3T^{2} $
	67	$ 1 + 109.T + 4.48e3T^{2} $
	71	$ 1 + 57.6iT - 5.04e3T^{2} $
	73	$ 1 - 36.4iT - 5.32e3T^{2} $
	79	$ 1 + 112. iT - 6.24e3T^{2} $
	83	$ 1 + 42.1iT - 6.88e3T^{2} $
	89	$ 1 + 101. iT - 7.92e3T^{2} $
	97	$ 1 + 148.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

−8.880936187258852914422570562575, −7.75712384515696864644364625728, −7.60694929217384005962279389192, −6.27505648374239069885864903578, −5.76287492638191319840758857489, −4.97352556743477247079280833106, −4.07073620359401983332177203161, −2.98609565278481083739004359863, −1.67831242269723937804314726734, −0.28726170888419316032073649813, 0.57420579503803524710756318746, 2.44460159999150654824575899481, 2.91489147949998145031141269550, 4.66536443601019925423089730102, 5.24192878314937981520649400240, 5.65817161082977355943066882016, 6.85008062860854971485002751811, 7.35528687176312660976998421070, 8.532157410240576321768836929528, 9.022734661904251334557901173186

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.5487537505\)
\(L(\frac12)\)	\(\approx\)	\(0.5487537505\)
\(L(2)\)		not available
\(L(1)\)		not available

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