LMFDB - L-function 2-3100-155.154-c2-0-22

Dirichlet series

L(s) = 1

− 1.25·3^-s + 6.21i·7^-s − 7.43·9^-s − 14.0i·11^-s − 19.3·13^-s − 28.3·17^-s − 12.6·19^-s − 7.77i·21^-s − 5.27·23^-s + 20.5·27^-s − 34.3i·29^-s + (23.4 + 20.2i)31^-s + 17.5i·33^-s − 34.6·37^-s + 24.1·39^-s + ⋯

L(s) = 1

− 0.417·3^-s + 0.888i·7^-s − 0.825·9^-s − 1.27i·11^-s − 1.48·13^-s − 1.66·17^-s − 0.665·19^-s − 0.370i·21^-s − 0.229·23^-s + 0.761·27^-s − 1.18i·29^-s + (0.755 + 0.654i)31^-s + 0.532i·33^-s − 0.935·37^-s + 0.619·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3100$ = $2^{2} \cdot 5^{2} \cdot 31$
Sign:	$0.923 - 0.383i$
Analytic conductor:	$84.4688$
Root analytic conductor:	$9.19069$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3100} (1549, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3100,\ (\ :1),\ 0.923 - 0.383i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.6019533749$
$L(\frac12)$	$\approx$	$0.6019533749$
$L(2)$		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 + (-23.4 - 20.2i)T $
good	3	$ 1 + 1.25T + 9T^{2} $
	7	$ 1 - 6.21iT - 49T^{2} $
	11	$ 1 + 14.0iT - 121T^{2} $
	13	$ 1 + 19.3T + 169T^{2} $
	17	$ 1 + 28.3T + 289T^{2} $
	19	$ 1 + 12.6T + 361T^{2} $
	23	$ 1 + 5.27T + 529T^{2} $
	29	$ 1 + 34.3iT - 841T^{2} $
	37	$ 1 + 34.6T + 1.36e3T^{2} $
	41	$ 1 + 59.5T + 1.68e3T^{2} $
	43	$ 1 + 1.25T + 1.84e3T^{2} $
	47	$ 1 + 22.1iT - 2.20e3T^{2} $
	53	$ 1 - 31.5T + 2.80e3T^{2} $
	59	$ 1 - 41.9T + 3.48e3T^{2} $
	61	$ 1 + 46.8iT - 3.72e3T^{2} $
	67	$ 1 - 45.1iT - 4.48e3T^{2} $
	71	$ 1 - 67.2T + 5.04e3T^{2} $
	73	$ 1 - 48.6T + 5.32e3T^{2} $
	79	$ 1 - 151. iT - 6.24e3T^{2} $
	83	$ 1 - 87.7T + 6.88e3T^{2} $
	89	$ 1 + 98.0iT - 7.92e3T^{2} $
	97	$ 1 - 100. iT - 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

−8.581499664632769771256623510612, −8.078360647999133114065009631970, −6.76351667462120082842087840298, −6.37925872380246108779930237610, −5.41944389450838918459596489095, −4.98886109409366248740367649592, −3.85680209431167460049694522420, −2.68608370115570421500660135541, −2.21332035611078013148038741946, −0.41451300848603453488783996075, 0.31396764679436622902269122789, 1.89537532635625249062313621524, 2.64297221713433702099927659281, 3.93168895011788558345112315427, 4.72968375568806673796266788243, 5.16573196831502254599662618201, 6.46384347283097500324726322001, 6.89610538007496034205098460614, 7.56770424615072533915119346727, 8.506281685103120101034304371188

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.6019533749\)
\(L(\frac12)\)	\(\approx\)	\(0.6019533749\)
\(L(2)\)		not available
\(L(1)\)		not available

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