LMFDB - L-function 2-4140-23.22-c2-0-71

Dirichlet series

L(s) = 1

− 2.23i·5^-s − 4.50i·7^-s − 16.7i·11^-s + 3.28·13^-s − 7.91i·17^-s + 5.02i·19^-s + (−9.32 − 21.0i)23^-s − 5.00·25^-s + 46.6·29^-s − 19.2·31^-s − 10.0·35^-s − 7.65i·37^-s + 44.7·41^-s − 33.2i·43^-s + 34.0·47^-s + ⋯

L(s) = 1

− 0.447i·5^-s − 0.643i·7^-s − 1.52i·11^-s + 0.252·13^-s − 0.465i·17^-s + 0.264i·19^-s + (−0.405 − 0.914i)23^-s − 0.200·25^-s + 1.60·29^-s − 0.622·31^-s − 0.287·35^-s − 0.206i·37^-s + 1.09·41^-s − 0.774i·43^-s + 0.724·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.405i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4140$ = $2^{2} \cdot 3^{2} \cdot 5 \cdot 23$
Sign:	$-0.914 + 0.405i$
Analytic conductor:	$112.806$
Root analytic conductor:	$10.6210$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4140} (2161, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4140,\ (\ :1),\ -0.914 + 0.405i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.704647151$
$L(\frac12)$	$\approx$	$1.704647151$
$L(2)$		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 $
	3	$ 1 $
	5	$ 1 + 2.23iT $
	23	$ 1 + (9.32 + 21.0i)T $
good	7	$ 1 + 4.50iT - 49T^{2} $
	11	$ 1 + 16.7iT - 121T^{2} $
	13	$ 1 - 3.28T + 169T^{2} $
	17	$ 1 + 7.91iT - 289T^{2} $
	19	$ 1 - 5.02iT - 361T^{2} $
	29	$ 1 - 46.6T + 841T^{2} $
	31	$ 1 + 19.2T + 961T^{2} $
	37	$ 1 + 7.65iT - 1.36e3T^{2} $
	41	$ 1 - 44.7T + 1.68e3T^{2} $
	43	$ 1 + 33.2iT - 1.84e3T^{2} $
	47	$ 1 - 34.0T + 2.20e3T^{2} $
	53	$ 1 + 55.1iT - 2.80e3T^{2} $
	59	$ 1 + 41.8T + 3.48e3T^{2} $
	61	$ 1 + 60.8iT - 3.72e3T^{2} $
	67	$ 1 + 99.3iT - 4.48e3T^{2} $
	71	$ 1 + 75.5T + 5.04e3T^{2} $
	73	$ 1 + 4.63T + 5.32e3T^{2} $
	79	$ 1 - 146. iT - 6.24e3T^{2} $
	83	$ 1 - 95.3iT - 6.88e3T^{2} $
	89	$ 1 + 18.9iT - 7.92e3T^{2} $
	97	$ 1 - 39.5iT - 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

−8.095502217585313489101599993703, −7.22503768903328462240323064705, −6.38482298090376222944813442479, −5.77836032781804936139571686516, −4.91362953078125014709662468680, −4.08011711405904955538048410812, −3.36108577105684533546031737500, −2.37949089964036947565694074962, −1.04533386965277277627854139937, −0.39811773218553735576850457807, 1.31819402873764804638778862827, 2.26148762371014358241575647223, 3.00019260381656232982975161684, 4.12463313179967869681606728457, 4.72287141579259414636913605342, 5.75900255903614885700598784742, 6.27637125190605878106224672549, 7.28765760529481046964431228396, 7.60309761089618326791301358745, 8.679568767410734176871747104686

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.704647151\)
\(L(\frac12)\)	\(\approx\)	\(1.704647151\)
\(L(2)\)		not available
\(L(1)\)		not available

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