LMFDB - L-function 2-1900-19.18-c2-0-38

Dirichlet series

L(s) = 1

− 0.284i·3^-s − 2.19·7^-s + 8.91·9^-s − 2.14·11^-s − 5.91i·13^-s + 21.3·17^-s + (−12.8 − 13.9i)19^-s + 0.624i·21^-s + 9.06·23^-s − 5.09i·27^-s + 33.2i·29^-s + 44.1i·31^-s + 0.610i·33^-s − 50.0i·37^-s − 1.68·39^-s + ⋯

L(s) = 1

− 0.0947i·3^-s − 0.313·7^-s + 0.991·9^-s − 0.195·11^-s − 0.454i·13^-s + 1.25·17^-s + (−0.676 − 0.736i)19^-s + 0.0297i·21^-s + 0.393·23^-s − 0.188i·27^-s + 1.14i·29^-s + 1.42i·31^-s + 0.0185i·33^-s − 1.35i·37^-s − 0.0430·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1900$ = $2^{2} \cdot 5^{2} \cdot 19$
Sign:	$0.676 + 0.736i$
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:	$(2,\ 1900,\ (\ :1),\ 0.676 + 0.736i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.025775919$
$L(\frac12)$	$\approx$	$2.025775919$
$L(2)$		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 + (12.8 + 13.9i)T $
good	3	$ 1 + 0.284iT - 9T^{2} $
	7	$ 1 + 2.19T + 49T^{2} $
	11	$ 1 + 2.14T + 121T^{2} $
	13	$ 1 + 5.91iT - 169T^{2} $
	17	$ 1 - 21.3T + 289T^{2} $
	23	$ 1 - 9.06T + 529T^{2} $
	29	$ 1 - 33.2iT - 841T^{2} $
	31	$ 1 - 44.1iT - 961T^{2} $
	37	$ 1 + 50.0iT - 1.36e3T^{2} $
	41	$ 1 + 61.1iT - 1.68e3T^{2} $
	43	$ 1 - 5.39T + 1.84e3T^{2} $
	47	$ 1 + 9.02T + 2.20e3T^{2} $
	53	$ 1 + 62.0iT - 2.80e3T^{2} $
	59	$ 1 - 37.5iT - 3.48e3T^{2} $
	61	$ 1 - 58.0T + 3.72e3T^{2} $
	67	$ 1 - 121. iT - 4.48e3T^{2} $
	71	$ 1 + 103. iT - 5.04e3T^{2} $
	73	$ 1 - 120.T + 5.32e3T^{2} $
	79	$ 1 - 57.1iT - 6.24e3T^{2} $
	83	$ 1 - 47.3T + 6.88e3T^{2} $
	89	$ 1 + 68.2iT - 7.92e3T^{2} $
	97	$ 1 + 22.7iT - 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

−8.938874581910942026506182854299, −8.146546708149026851270507228783, −7.17263367925102042621294232766, −6.82006121568636522751468840977, −5.60897273610705376953771875232, −4.95991568513967871977795553128, −3.85835640085503068762471224153, −3.04797536858582632417516719339, −1.80677950441604345759330978910, −0.63815478365142931429067688279, 0.991184258591618826208716825596, 2.12323920080147586297425203861, 3.33826895110033196499464626942, 4.18123152178813316138069251631, 4.98888497479738958944813838624, 6.09960055330842283949726323945, 6.64474723612810079305767305472, 7.78387418135545966167993022224, 8.084831002316615825819204075271, 9.439328318042697880667382755614

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.025775919\)
\(L(\frac12)\)	\(\approx\)	\(2.025775919\)
\(L(2)\)		not available
\(L(1)\)		not available

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