LMFDB - L-function 2-1900-19.18-c2-0-3

Dirichlet series

L(s) = 1

− 5.61i·3^-s + 7.03·7^-s − 22.5·9^-s − 1.02·11^-s + 11.3i·13^-s − 16.5·17^-s + (9.54 − 16.4i)19^-s − 39.4i·21^-s − 35.1·23^-s + 75.9i·27^-s − 6.63i·29^-s + 25.3i·31^-s + 5.73i·33^-s + 47.8i·37^-s + 63.7·39^-s + ⋯

L(s) = 1

− 1.87i·3^-s + 1.00·7^-s − 2.50·9^-s − 0.0928·11^-s + 0.872i·13^-s − 0.975·17^-s + (0.502 − 0.864i)19^-s − 1.88i·21^-s − 1.52·23^-s + 2.81i·27^-s − 0.228i·29^-s + 0.818i·31^-s + 0.173i·33^-s + 1.29i·37^-s + 1.63·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1900$ = $2^{2} \cdot 5^{2} \cdot 19$
Sign:	$0.502 - 0.864i$
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:	$(2,\ 1900,\ (\ :1),\ 0.502 - 0.864i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.4070410187$
$L(\frac12)$	$\approx$	$0.4070410187$
$L(2)$		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 + (-9.54 + 16.4i)T $
good	3	$ 1 + 5.61iT - 9T^{2} $
	7	$ 1 - 7.03T + 49T^{2} $
	11	$ 1 + 1.02T + 121T^{2} $
	13	$ 1 - 11.3iT - 169T^{2} $
	17	$ 1 + 16.5T + 289T^{2} $
	23	$ 1 + 35.1T + 529T^{2} $
	29	$ 1 + 6.63iT - 841T^{2} $
	31	$ 1 - 25.3iT - 961T^{2} $
	37	$ 1 - 47.8iT - 1.36e3T^{2} $
	41	$ 1 - 12.1iT - 1.68e3T^{2} $
	43	$ 1 - 50.0T + 1.84e3T^{2} $
	47	$ 1 + 78.3T + 2.20e3T^{2} $
	53	$ 1 + 45.0iT - 2.80e3T^{2} $
	59	$ 1 - 9.44iT - 3.48e3T^{2} $
	61	$ 1 + 43.1T + 3.72e3T^{2} $
	67	$ 1 - 112. iT - 4.48e3T^{2} $
	71	$ 1 - 40.3iT - 5.04e3T^{2} $
	73	$ 1 + 45.4T + 5.32e3T^{2} $
	79	$ 1 - 144. iT - 6.24e3T^{2} $
	83	$ 1 + 56.8T + 6.88e3T^{2} $
	89	$ 1 + 81.8iT - 7.92e3T^{2} $
	97	$ 1 + 103. iT - 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

−8.721669380573608878538626059540, −8.305292005043650422394668657654, −7.54454207984562644819955220279, −6.83857777159052439784089476143, −6.27568973683257851554038605666, −5.27794117505034768807069655407, −4.34386346393553438333928924704, −2.82772344368122806498562725892, −1.96571668935278188111139857162, −1.26962603462727380252331800406, 0.10063367829220773898271280290, 2.04228749795482789647504606258, 3.21738834226985476781283267101, 4.08564417514219099796722456792, 4.67718596996736058008843095694, 5.52301712080862461594337353780, 6.09124989626095524678955155039, 7.74970298189430172832600072113, 8.166387963080107605506892832883, 9.105295404491284118115718496332

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.4070410187\)
\(L(\frac12)\)	\(\approx\)	\(0.4070410187\)
\(L(2)\)		not available
\(L(1)\)		not available

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