LMFDB - L-function 2-1200-4.3-c2-0-20

Dirichlet series

L(s) = 1

− 1.73i·3^-s + 1.73i·7^-s − 2.99·9^-s + 3.46i·11^-s + 11·13^-s − 6·17^-s − 19.0i·19^-s + 2.99·21^-s + 17.3i·23^-s + 5.19i·27^-s + 30·29^-s + 5.19i·31^-s + 5.99·33^-s − 14·37^-s − 19.0i·39^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.247i·7^-s − 0.333·9^-s + 0.314i·11^-s + 0.846·13^-s − 0.352·17^-s − 1.00i·19^-s + 0.142·21^-s + 0.753i·23^-s + 0.192i·27^-s + 1.03·29^-s + 0.167i·31^-s + 0.181·33^-s − 0.378·37^-s − 0.488i·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1200$ = $2^{4} \cdot 3 \cdot 5^{2}$
Sign:	$0.866 + 0.5i$
Analytic conductor:	$32.6976$
Root analytic conductor:	$5.71818$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1200} (751, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1200,\ (\ :1),\ 0.866 + 0.5i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.939818782$
$L(\frac12)$	$\approx$	$1.939818782$
$L(2)$		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 + 1.73iT $
	5	$ 1 $
good	7	$ 1 - 1.73iT - 49T^{2} $
	11	$ 1 - 3.46iT - 121T^{2} $
	13	$ 1 - 11T + 169T^{2} $
	17	$ 1 + 6T + 289T^{2} $
	19	$ 1 + 19.0iT - 361T^{2} $
	23	$ 1 - 17.3iT - 529T^{2} $
	29	$ 1 - 30T + 841T^{2} $
	31	$ 1 - 5.19iT - 961T^{2} $
	37	$ 1 + 14T + 1.36e3T^{2} $
	41	$ 1 - 36T + 1.68e3T^{2} $
	43	$ 1 + 5.19iT - 1.84e3T^{2} $
	47	$ 1 - 45.0iT - 2.20e3T^{2} $
	53	$ 1 - 72T + 2.80e3T^{2} $
	59	$ 1 + 38.1iT - 3.48e3T^{2} $
	61	$ 1 - 35T + 3.72e3T^{2} $
	67	$ 1 - 29.4iT - 4.48e3T^{2} $
	71	$ 1 + 90.0iT - 5.04e3T^{2} $
	73	$ 1 - 62T + 5.32e3T^{2} $
	79	$ 1 + 76.2iT - 6.24e3T^{2} $
	83	$ 1 + 72.7iT - 6.88e3T^{2} $
	89	$ 1 - 144T + 7.92e3T^{2} $
	97	$ 1 + 181T + 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

−9.277024016972769991418258806634, −8.721907400478187345439096459074, −7.81626809656344545384298737569, −6.98199608166026379897140855602, −6.24678899701115415248065522447, −5.33450687367150228854356641634, −4.30198377991673396797119295691, −3.11819967364854970658911582103, −2.06762422047041553387467850162, −0.813850784254123610333572575816, 0.881064909224726689362284050604, 2.41461185538525208639719387484, 3.62600218124160869113241599090, 4.28471151223432583439968806334, 5.41334602129417190832324176100, 6.19446713217774802696492460360, 7.09648619318535740963081717066, 8.284616755113088247610197274733, 8.673899084712703896135718132040, 9.727561090335730315227702430514

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 $\F_{q}$
Belyi maps

\(L(\frac{3}{2})\)	\(\approx\)	\(1.939818782\)
\(L(\frac12)\)	\(\approx\)	\(1.939818782\)
\(L(2)\)		not available
\(L(1)\)		not available

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