LMFDB - L-function 2-1143-3.2-c2-0-46

Dirichlet series

L(s) = 1

− 3.21i·2^-s − 6.30·4^-s − 0.600i·5^-s + 12.4·7^-s + 7.41i·8^-s − 1.92·10^-s + 13.0i·11^-s − 16.5·13^-s − 40.0i·14^-s − 1.43·16^-s − 8.91i·17^-s + 36.7·19^-s + 3.79i·20^-s + 41.8·22^-s + 24.0i·23^-s + ⋯

L(s) = 1

− 1.60i·2^-s − 1.57·4^-s − 0.120i·5^-s + 1.78·7^-s + 0.926i·8^-s − 0.192·10^-s + 1.18i·11^-s − 1.27·13^-s − 2.86i·14^-s − 0.0894·16^-s − 0.524i·17^-s + 1.93·19^-s + 0.189i·20^-s + 1.90·22^-s + 1.04i·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1143$ = $3^{2} \cdot 127$
Sign:	$-0.577 + 0.816i$
Analytic conductor:	$31.1444$
Root analytic conductor:	$5.58072$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1143} (890, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1143,\ (\ :1),\ -0.577 + 0.816i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.233262188$
$L(\frac12)$	$\approx$	$2.233262188$
$L(2)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 $
	127	$ 1 + 11.2T $
good	2	$ 1 + 3.21iT - 4T^{2} $
	5	$ 1 + 0.600iT - 25T^{2} $
	7	$ 1 - 12.4T + 49T^{2} $
	11	$ 1 - 13.0iT - 121T^{2} $
	13	$ 1 + 16.5T + 169T^{2} $
	17	$ 1 + 8.91iT - 289T^{2} $
	19	$ 1 - 36.7T + 361T^{2} $
	23	$ 1 - 24.0iT - 529T^{2} $
	29	$ 1 + 31.9iT - 841T^{2} $
	31	$ 1 + 22.1T + 961T^{2} $
	37	$ 1 - 51.0T + 1.36e3T^{2} $
	41	$ 1 + 20.9iT - 1.68e3T^{2} $
	43	$ 1 - 23.4T + 1.84e3T^{2} $
	47	$ 1 + 36.6iT - 2.20e3T^{2} $
	53	$ 1 - 6.11iT - 2.80e3T^{2} $
	59	$ 1 + 48.2iT - 3.48e3T^{2} $
	61	$ 1 - 103.T + 3.72e3T^{2} $
	67	$ 1 - 39.6T + 4.48e3T^{2} $
	71	$ 1 - 19.8iT - 5.04e3T^{2} $
	73	$ 1 - 13.6T + 5.32e3T^{2} $
	79	$ 1 + 81.4T + 6.24e3T^{2} $
	83	$ 1 + 64.4iT - 6.88e3T^{2} $
	89	$ 1 + 106. iT - 7.92e3T^{2} $
	97	$ 1 - 97.3T + 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

−9.717111490876056578132579303159, −8.856977393659601981805833107221, −7.61287374602839014535820813009, −7.31089156984711351818923167591, −5.23880340384096019019341265849, −4.90058753493133402530580181447, −3.98107091395112166970319372975, −2.64592429008420212497853238285, −1.88919754923674361319105009469, −0.897162992312977664162723486102, 1.02481610438606003532931453160, 2.69789835971449063008871079577, 4.26737705267614867234943006176, 5.17642943982057900259810363948, 5.50278657709631160870292123707, 6.70771747482606091125127121584, 7.50723649108180257850005384137, 8.038683933753300287104771600999, 8.694868229348026958303353663313, 9.522017823242879925848091486903

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.233262188\)
\(L(\frac12)\)	\(\approx\)	\(2.233262188\)
\(L(2)\)		not available
\(L(1)\)		not available

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