LMFDB - L-function 2-60e2-15.14-c2-0-2

Dirichlet series

L(s) = 1

− 4i·7^-s + 16.9i·11^-s + 8i·13^-s + 12.7·17^-s − 16·19^-s + 16.9·23^-s + 4.24i·29^-s − 44·31^-s + 34i·37^-s − 46.6i·41^-s + 40i·43^-s − 84.8·47^-s + 33·49^-s + 38.1·53^-s − 33.9i·59^-s + ⋯

L(s) = 1

− 0.571i·7^-s + 1.54i·11^-s + 0.615i·13^-s + 0.748·17^-s − 0.842·19^-s + 0.737·23^-s + 0.146i·29^-s − 1.41·31^-s + 0.918i·37^-s − 1.13i·41^-s + 0.930i·43^-s − 1.80·47^-s + 0.673·49^-s + 0.720·53^-s − 0.575i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3600$ = $2^{4} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.988 - 0.151i$
Analytic conductor:	$98.0928$
Root analytic conductor:	$9.90418$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3600} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3600,\ (\ :1),\ -0.988 - 0.151i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.4913341999$
$L(\frac12)$	$\approx$	$0.4913341999$
$L(2)$		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 $
good	7	$ 1 + 4iT - 49T^{2} $
	11	$ 1 - 16.9iT - 121T^{2} $
	13	$ 1 - 8iT - 169T^{2} $
	17	$ 1 - 12.7T + 289T^{2} $
	19	$ 1 + 16T + 361T^{2} $
	23	$ 1 - 16.9T + 529T^{2} $
	29	$ 1 - 4.24iT - 841T^{2} $
	31	$ 1 + 44T + 961T^{2} $
	37	$ 1 - 34iT - 1.36e3T^{2} $
	41	$ 1 + 46.6iT - 1.68e3T^{2} $
	43	$ 1 - 40iT - 1.84e3T^{2} $
	47	$ 1 + 84.8T + 2.20e3T^{2} $
	53	$ 1 - 38.1T + 2.80e3T^{2} $
	59	$ 1 + 33.9iT - 3.48e3T^{2} $
	61	$ 1 - 50T + 3.72e3T^{2} $
	67	$ 1 - 8iT - 4.48e3T^{2} $
	71	$ 1 + 50.9iT - 5.04e3T^{2} $
	73	$ 1 + 16iT - 5.32e3T^{2} $
	79	$ 1 + 76T + 6.24e3T^{2} $
	83	$ 1 + 118.T + 6.88e3T^{2} $
	89	$ 1 - 12.7iT - 7.92e3T^{2} $
	97	$ 1 + 176iT - 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

−8.749067544752064992222070231560, −7.910271589716333042802374754265, −7.07295009022949537103456283352, −6.83184783912219683337427169127, −5.69208431744138402385693249313, −4.81228236759558101277472382038, −4.21966178394235150047140077182, −3.33179983992862492140792891550, −2.15188237022645097933642209824, −1.36720093869094286066583849534, 0.10689940642333593782968600210, 1.21062164738229565714595689696, 2.47684030591737260637085311734, 3.26201343447385369208709525195, 4.00845810303439718205774949314, 5.31721790804247099144835457599, 5.62081713941664506806379264578, 6.44714174778300979083622337958, 7.33292875222890968667252459432, 8.208982584686578266918751181098

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.4913341999\)
\(L(\frac12)\)	\(\approx\)	\(0.4913341999\)
\(L(2)\)		not available
\(L(1)\)		not available

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