LMFDB - L-function 2-3100-31.30-c2-0-5

Dirichlet series

L(s) = 1

− 0.767i·3^-s − 9.99·7^-s + 8.41·9^-s + 13.1i·11^-s + 22.5i·13^-s + 10.8i·17^-s + 1.94·19^-s + 7.67i·21^-s − 9.36i·23^-s − 13.3i·27^-s + 4.91i·29^-s + (−30.6 + 4.71i)31^-s + 10.0·33^-s + 2.70i·37^-s + 17.3·39^-s + ⋯

L(s) = 1

− 0.255i·3^-s − 1.42·7^-s + 0.934·9^-s + 1.19i·11^-s + 1.73i·13^-s + 0.635i·17^-s + 0.102·19^-s + 0.365i·21^-s − 0.407i·23^-s − 0.494i·27^-s + 0.169i·29^-s + (−0.988 + 0.152i)31^-s + 0.305·33^-s + 0.0732i·37^-s + 0.443·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.152i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3100$ = $2^{2} \cdot 5^{2} \cdot 31$
Sign:	$-0.988 + 0.152i$
Analytic conductor:	$84.4688$
Root analytic conductor:	$9.19069$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3100} (1301, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3100,\ (\ :1),\ -0.988 + 0.152i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.4357559864$
$L(\frac12)$	$\approx$	$0.4357559864$
$L(2)$		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 $
	31	$ 1 + (30.6 - 4.71i)T $
good	3	$ 1 + 0.767iT - 9T^{2} $
	7	$ 1 + 9.99T + 49T^{2} $
	11	$ 1 - 13.1iT - 121T^{2} $
	13	$ 1 - 22.5iT - 169T^{2} $
	17	$ 1 - 10.8iT - 289T^{2} $
	19	$ 1 - 1.94T + 361T^{2} $
	23	$ 1 + 9.36iT - 529T^{2} $
	29	$ 1 - 4.91iT - 841T^{2} $
	37	$ 1 - 2.70iT - 1.36e3T^{2} $
	41	$ 1 - 16.3T + 1.68e3T^{2} $
	43	$ 1 - 69.4iT - 1.84e3T^{2} $
	47	$ 1 + 67.1T + 2.20e3T^{2} $
	53	$ 1 + 86.4iT - 2.80e3T^{2} $
	59	$ 1 - 59.8T + 3.48e3T^{2} $
	61	$ 1 + 41.0iT - 3.72e3T^{2} $
	67	$ 1 - 17.5T + 4.48e3T^{2} $
	71	$ 1 + 2.94T + 5.04e3T^{2} $
	73	$ 1 + 91.1iT - 5.32e3T^{2} $
	79	$ 1 - 89.4iT - 6.24e3T^{2} $
	83	$ 1 + 30.9iT - 6.88e3T^{2} $
	89	$ 1 + 94.1iT - 7.92e3T^{2} $
	97	$ 1 - 129.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

−9.156295887292669755771644655975, −8.096646356750175158532410519716, −7.13805186334856369428915113220, −6.72544220758804041870200141366, −6.24690668272019276703776702171, −4.92778207307051131285645777766, −4.20103952212838216177736370124, −3.48568071144603644924803452826, −2.22267291900359412702923027135, −1.49882106739268203766880264266, 0.10891253957667733887404853629, 1.00951375747309168599132148548, 2.64817927282460382871548530283, 3.36604385483366811533020836826, 3.93586233779321882914146881107, 5.27830104400668611610986415391, 5.74077971561216760337285949734, 6.63615419755151525744527003672, 7.35734666756130588775999284720, 8.094484804301379878205134115931

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.4357559864\)
\(L(\frac12)\)	\(\approx\)	\(0.4357559864\)
\(L(2)\)		not available
\(L(1)\)		not available

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