LMFDB - L-function 2-1900-95.94-c2-0-32

Dirichlet series

L(s) = 1

− 3.14·3^-s − 12.2i·7^-s + 0.877·9^-s + 0.636·11^-s + 19.9·13^-s + 20.7i·17^-s + (4.37 − 18.4i)19^-s + 38.4i·21^-s + 6.16i·23^-s + 25.5·27^-s + 11.9i·29^-s + 47.7i·31^-s − 1.99·33^-s + 19.1·37^-s − 62.5·39^-s + ⋯

L(s) = 1

− 1.04·3^-s − 1.74i·7^-s + 0.0975·9^-s + 0.0578·11^-s + 1.53·13^-s + 1.21i·17^-s + (0.230 − 0.973i)19^-s + 1.82i·21^-s + 0.267i·23^-s + 0.945·27^-s + 0.413i·29^-s + 1.54i·31^-s − 0.0605·33^-s + 0.518·37^-s − 1.60·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1900$ = $2^{2} \cdot 5^{2} \cdot 19$
Sign:	$0.767 + 0.641i$
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:	$(2,\ 1900,\ (\ :1),\ 0.767 + 0.641i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.359697119$
$L(\frac12)$	$\approx$	$1.359697119$
$L(2)$		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 + (-4.37 + 18.4i)T $
good	3	$ 1 + 3.14T + 9T^{2} $
	7	$ 1 + 12.2iT - 49T^{2} $
	11	$ 1 - 0.636T + 121T^{2} $
	13	$ 1 - 19.9T + 169T^{2} $
	17	$ 1 - 20.7iT - 289T^{2} $
	23	$ 1 - 6.16iT - 529T^{2} $
	29	$ 1 - 11.9iT - 841T^{2} $
	31	$ 1 - 47.7iT - 961T^{2} $
	37	$ 1 - 19.1T + 1.36e3T^{2} $
	41	$ 1 - 22.7iT - 1.68e3T^{2} $
	43	$ 1 - 34.1iT - 1.84e3T^{2} $
	47	$ 1 + 75.4iT - 2.20e3T^{2} $
	53	$ 1 - 87.2T + 2.80e3T^{2} $
	59	$ 1 + 40.7iT - 3.48e3T^{2} $
	61	$ 1 + 10.2T + 3.72e3T^{2} $
	67	$ 1 + 16.8T + 4.48e3T^{2} $
	71	$ 1 - 23.2iT - 5.04e3T^{2} $
	73	$ 1 - 103. iT - 5.32e3T^{2} $
	79	$ 1 + 123. iT - 6.24e3T^{2} $
	83	$ 1 - 82.9iT - 6.88e3T^{2} $
	89	$ 1 - 34.0iT - 7.92e3T^{2} $
	97	$ 1 - 66.0T + 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

−8.821756182990355076711030002061, −8.190426950134494561049259394554, −7.08555350646922553762731601736, −6.61228073370790375484938790007, −5.82243308787681818807577871163, −4.88625656764805256053586223581, −4.02184679968744500660661242428, −3.28781948634872067622649473129, −1.40994297280469969171990295051, −0.67572058442313381525310734109, 0.73309216823327500712445605194, 2.13564058589037326860745292261, 3.12559645860733514572823828965, 4.35097127906593212333048686834, 5.42183565456113112223500190294, 5.88767202859963161970019681469, 6.31344636172424962712903322607, 7.55514424016694326405184908577, 8.504559880874693564540479997294, 9.035864456314700875790525610765

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.359697119\)
\(L(\frac12)\)	\(\approx\)	\(1.359697119\)
\(L(2)\)		not available
\(L(1)\)		not available

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