LMFDB - L-function 2-3100-31.30-c2-0-65

Dirichlet series

L(s) = 1

− 2.65i·3^-s + 0.895·7^-s + 1.96·9^-s + 1.97i·11^-s − 4.58i·13^-s − 13.5i·17^-s + 13.3·19^-s − 2.37i·21^-s + 8.61i·23^-s − 29.0i·27^-s + 15.3i·29^-s + (5.19 + 30.5i)31^-s + 5.23·33^-s + 47.2i·37^-s − 12.1·39^-s + ⋯

L(s) = 1

− 0.883i·3^-s + 0.127·7^-s + 0.218·9^-s + 0.179i·11^-s − 0.352i·13^-s − 0.796i·17^-s + 0.704·19^-s − 0.113i·21^-s + 0.374i·23^-s − 1.07i·27^-s + 0.530i·29^-s + (0.167 + 0.985i)31^-s + 0.158·33^-s + 1.27i·37^-s − 0.311·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.167 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3100$ = $2^{2} \cdot 5^{2} \cdot 31$
Sign:	$0.167 + 0.985i$
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:	$(2,\ 3100,\ (\ :1),\ 0.167 + 0.985i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.326623311$
$L(\frac12)$	$\approx$	$2.326623311$
$L(2)$		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 + (-5.19 - 30.5i)T $
good	3	$ 1 + 2.65iT - 9T^{2} $
	7	$ 1 - 0.895T + 49T^{2} $
	11	$ 1 - 1.97iT - 121T^{2} $
	13	$ 1 + 4.58iT - 169T^{2} $
	17	$ 1 + 13.5iT - 289T^{2} $
	19	$ 1 - 13.3T + 361T^{2} $
	23	$ 1 - 8.61iT - 529T^{2} $
	29	$ 1 - 15.3iT - 841T^{2} $
	37	$ 1 - 47.2iT - 1.36e3T^{2} $
	41	$ 1 - 75.1T + 1.68e3T^{2} $
	43	$ 1 + 62.5iT - 1.84e3T^{2} $
	47	$ 1 - 82.9T + 2.20e3T^{2} $
	53	$ 1 + 60.1iT - 2.80e3T^{2} $
	59	$ 1 + 62.3T + 3.48e3T^{2} $
	61	$ 1 + 10.2iT - 3.72e3T^{2} $
	67	$ 1 + 29.8T + 4.48e3T^{2} $
	71	$ 1 - 38.0T + 5.04e3T^{2} $
	73	$ 1 - 2.30iT - 5.32e3T^{2} $
	79	$ 1 - 110. iT - 6.24e3T^{2} $
	83	$ 1 + 26.9iT - 6.88e3T^{2} $
	89	$ 1 + 55.6iT - 7.92e3T^{2} $
	97	$ 1 - 13.5T + 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

−8.206527312030231799191495858888, −7.43146537885486137022744048327, −7.07584540001482867672244502880, −6.20230349933521478593260179269, −5.32338033780589287102652721213, −4.57529518331708104998311251107, −3.45814705152294826429672430243, −2.54311232482411135059669849951, −1.51681074350022151883952764335, −0.65821417597100936394006908643, 0.929686347897694807002398143619, 2.15999754226588676084034282987, 3.24102828368363984011974338367, 4.20390004520333770298638534702, 4.53626555069398941604918049355, 5.69803246368368902426215273363, 6.21142977848522078352489030970, 7.39141334534485552254780457057, 7.85131047580986359391604939382, 8.952561736800682774056028971369

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.326623311\)
\(L(\frac12)\)	\(\approx\)	\(2.326623311\)
\(L(2)\)		not available
\(L(1)\)		not available

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