LMFDB - L-function 2-1400-5.4-c3-0-30

Dirichlet series

L(s) = 1

− 9.69i·3^-s − 7i·7^-s − 66.9·9^-s − 8.69·11^-s + 66.1i·13^-s − 1.95i·17^-s + 109.·19^-s − 67.8·21^-s + 152. i·23^-s + 387. i·27^-s + 299.·29^-s − 142.·31^-s + 84.2i·33^-s − 318. i·37^-s + 641.·39^-s + ⋯

L(s) = 1

− 1.86i·3^-s − 0.377i·7^-s − 2.48·9^-s − 0.238·11^-s + 1.41i·13^-s − 0.0279i·17^-s + 1.32·19^-s − 0.705·21^-s + 1.38i·23^-s + 2.76i·27^-s + 1.91·29^-s − 0.824·31^-s + 0.444i·33^-s − 1.41i·37^-s + 2.63·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1400$ = $2^{3} \cdot 5^{2} \cdot 7$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$82.6026$
Root analytic conductor:	$9.08860$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1400} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1400,\ (\ :3/2),\ 0.447 + 0.894i)$

Particular Values

$L(2)$	$\approx$	$1.846801223$
$L(\frac12)$	$\approx$	$1.846801223$
$L(\frac{5}{2})$		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 $
	5	$ 1 $
	7	$ 1 + 7iT $
good	3	$ 1 + 9.69iT - 27T^{2} $
	11	$ 1 + 8.69T + 1.33e3T^{2} $
	13	$ 1 - 66.1iT - 2.19e3T^{2} $
	17	$ 1 + 1.95iT - 4.91e3T^{2} $
	19	$ 1 - 109.T + 6.85e3T^{2} $
	23	$ 1 - 152. iT - 1.21e4T^{2} $
	29	$ 1 - 299.T + 2.43e4T^{2} $
	31	$ 1 + 142.T + 2.97e4T^{2} $
	37	$ 1 + 318. iT - 5.06e4T^{2} $
	41	$ 1 + 54.6T + 6.89e4T^{2} $
	43	$ 1 + 98.7iT - 7.95e4T^{2} $
	47	$ 1 - 479. iT - 1.03e5T^{2} $
	53	$ 1 + 81.4iT - 1.48e5T^{2} $
	59	$ 1 - 651.T + 2.05e5T^{2} $
	61	$ 1 - 422.T + 2.26e5T^{2} $
	67	$ 1 - 325. iT - 3.00e5T^{2} $
	71	$ 1 - 501.T + 3.57e5T^{2} $
	73	$ 1 + 839. iT - 3.89e5T^{2} $
	79	$ 1 + 401.T + 4.93e5T^{2} $
	83	$ 1 + 363. iT - 5.71e5T^{2} $
	89	$ 1 + 514.T + 7.04e5T^{2} $
	97	$ 1 - 1.22e3iT - 9.12e5T^{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

−8.900794040808505164137726569003, −7.987848047877640835066410265455, −7.30442670388896300403063097504, −6.85785882262452097010627810961, −5.96893618194478487437259552365, −5.10273578681955056975479540630, −3.68463240442141048186566397110, −2.56534617947940068938261321180, −1.61547476316664241725605203464, −0.798407358131982002676912277793, 0.58379087599645976833308322536, 2.71717980158731758251328880418, 3.23404153586739173811453191441, 4.30564635742914747353174194393, 5.18862228859962004255552119458, 5.56433520466981001207014358034, 6.75696892050554563953326063547, 8.284883348396994270761458357632, 8.439711171565422378446141600959, 9.588947949657646055390530209826

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.846801223\)
\(L(\frac12)\)	\(\approx\)	\(1.846801223\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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