LMFDB - L-function 2-1150-115.114-c2-0-29

Dirichlet series

L(s) = 1

− 1.41i·2^-s + 4.30i·3^-s − 2.00·4^-s + 6.09·6^-s − 1.47·7^-s + 2.82i·8^-s − 9.55·9^-s − 6.04i·11^-s − 8.61i·12^-s − 5.21i·13^-s + 2.08i·14^-s + 4.00·16^-s + 15.7·17^-s + 13.5i·18^-s − 4.82i·19^-s + ⋯

L(s) = 1

− 0.707i·2^-s + 1.43i·3^-s − 0.500·4^-s + 1.01·6^-s − 0.210·7^-s + 0.353i·8^-s − 1.06·9^-s − 0.549i·11^-s − 0.717i·12^-s − 0.401i·13^-s + 0.149i·14^-s + 0.250·16^-s + 0.923·17^-s + 0.750i·18^-s − 0.254i·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1150$ = $2 \cdot 5^{2} \cdot 23$
Sign:	$0.838 - 0.544i$
Analytic conductor:	$31.3352$
Root analytic conductor:	$5.59778$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1150} (1149, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1150,\ (\ :1),\ 0.838 - 0.544i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.755816340$
$L(\frac12)$	$\approx$	$1.755816340$
$L(2)$		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 + 1.41iT $
	5	$ 1 $
	23	$ 1 + (-22.8 + 2.58i)T $
good	3	$ 1 - 4.30iT - 9T^{2} $
	7	$ 1 + 1.47T + 49T^{2} $
	11	$ 1 + 6.04iT - 121T^{2} $
	13	$ 1 + 5.21iT - 169T^{2} $
	17	$ 1 - 15.7T + 289T^{2} $
	19	$ 1 + 4.82iT - 361T^{2} $
	29	$ 1 - 23.4T + 841T^{2} $
	31	$ 1 + 20.4T + 961T^{2} $
	37	$ 1 - 15.5T + 1.36e3T^{2} $
	41	$ 1 - 20.3T + 1.68e3T^{2} $
	43	$ 1 - 38.1T + 1.84e3T^{2} $
	47	$ 1 - 13.8iT - 2.20e3T^{2} $
	53	$ 1 - 38.2T + 2.80e3T^{2} $
	59	$ 1 - 33.5T + 3.48e3T^{2} $
	61	$ 1 - 100. iT - 3.72e3T^{2} $
	67	$ 1 + 32.4T + 4.48e3T^{2} $
	71	$ 1 + 24.1T + 5.04e3T^{2} $
	73	$ 1 - 15.1iT - 5.32e3T^{2} $
	79	$ 1 - 11.2iT - 6.24e3T^{2} $
	83	$ 1 + 44.1T + 6.88e3T^{2} $
	89	$ 1 - 111. iT - 7.92e3T^{2} $
	97	$ 1 - 154.T + 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

−9.803338988517456263396039311845, −9.130407525466578194728299336777, −8.444742329430853430504063342480, −7.33562697158015491100223872130, −5.94283133494409441463582293057, −5.18390946426112308238282731296, −4.33227561170211559664928916417, −3.44391055193264516023494165818, −2.75161950355140463731594996656, −0.908118163137080723059495938214, 0.74106533848062232479730064581, 1.87287637805058473189843859306, 3.16728410646957095556690041233, 4.50442413202142724968965003290, 5.59162523019119945893701257519, 6.38137530435638034228163317930, 7.12521575209551597693618457759, 7.61964009696532264884671778754, 8.440532402365879102861384506507, 9.330982052675228841789371791060

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.755816340\)
\(L(\frac12)\)	\(\approx\)	\(1.755816340\)
\(L(2)\)		not available
\(L(1)\)		not available

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