LMFDB - L-function 2-3100-31.30-c2-0-42

Dirichlet series

L(s) = 1

+ 0.662i·3^-s + 8.05·7^-s + 8.56·9^-s + 14.8i·11^-s − 7.12i·13^-s + 26.3i·17^-s + 10.7·19^-s + 5.32i·21^-s − 6.66i·23^-s + 11.6i·27^-s − 5.50i·29^-s + (0.544 + 30.9i)31^-s − 9.81·33^-s + 30.4i·37^-s + 4.71·39^-s + ⋯

L(s) = 1

+ 0.220i·3^-s + 1.15·7^-s + 0.951·9^-s + 1.34i·11^-s − 0.548i·13^-s + 1.55i·17^-s + 0.565·19^-s + 0.253i·21^-s − 0.289i·23^-s + 0.430i·27^-s − 0.189i·29^-s + (0.0175 + 0.999i)31^-s − 0.297·33^-s + 0.822i·37^-s + 0.120·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0175 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0175 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3100$ = $2^{2} \cdot 5^{2} \cdot 31$
Sign:	$-0.0175 - 0.999i$
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:	$(2,\ 3100,\ (\ :1),\ -0.0175 - 0.999i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.601146797$
$L(\frac12)$	$\approx$	$2.601146797$
$L(2)$		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 + (-0.544 - 30.9i)T $
good	3	$ 1 - 0.662iT - 9T^{2} $
	7	$ 1 - 8.05T + 49T^{2} $
	11	$ 1 - 14.8iT - 121T^{2} $
	13	$ 1 + 7.12iT - 169T^{2} $
	17	$ 1 - 26.3iT - 289T^{2} $
	19	$ 1 - 10.7T + 361T^{2} $
	23	$ 1 + 6.66iT - 529T^{2} $
	29	$ 1 + 5.50iT - 841T^{2} $
	37	$ 1 - 30.4iT - 1.36e3T^{2} $
	41	$ 1 - 27.2T + 1.68e3T^{2} $
	43	$ 1 - 34.3iT - 1.84e3T^{2} $
	47	$ 1 + 22.6T + 2.20e3T^{2} $
	53	$ 1 + 42.2iT - 2.80e3T^{2} $
	59	$ 1 + 55.6T + 3.48e3T^{2} $
	61	$ 1 - 100. iT - 3.72e3T^{2} $
	67	$ 1 + 21.3T + 4.48e3T^{2} $
	71	$ 1 + 78.2T + 5.04e3T^{2} $
	73	$ 1 - 11.2iT - 5.32e3T^{2} $
	79	$ 1 + 28.9iT - 6.24e3T^{2} $
	83	$ 1 + 123. iT - 6.88e3T^{2} $
	89	$ 1 + 78.1iT - 7.92e3T^{2} $
	97	$ 1 + 24.3T + 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

−8.617558757808813169432600532233, −7.895083827935252162484971914800, −7.37205066775352640366629041643, −6.52570517908701625890568987616, −5.54462096430032439541711490042, −4.61086174251354522148279755054, −4.35109670560759442740357077337, −3.16089135608979449048576136593, −1.84193544965252666877390217267, −1.33118388865857673044115027998, 0.59930102473568939923636462623, 1.49444973363550129078258068137, 2.52661320173741433179472954036, 3.61548886087877039631513980774, 4.51809919345375170118483883863, 5.21074679686337617833201081198, 6.02521722456167644483665507549, 7.00192240482363075679702397742, 7.60436889656561673114920431010, 8.189026192129618391336373420423

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.601146797\)
\(L(\frac12)\)	\(\approx\)	\(2.601146797\)
\(L(2)\)		not available
\(L(1)\)		not available

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