LMFDB - L-function 2-1350-5.4-c3-0-10

Dirichlet series

L(s) = 1

− 2i·2^-s − 4·4^-s − 13i·7^-s + 8i·8^-s + 30·11^-s + 61i·13^-s − 26·14^-s + 16·16^-s − 12i·17^-s + 49·19^-s − 60i·22^-s + 18i·23^-s + 122·26^-s + 52i·28^-s − 186·29^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s − 0.701i·7^-s + 0.353i·8^-s + 0.822·11^-s + 1.30i·13^-s − 0.496·14^-s + 0.250·16^-s − 0.171i·17^-s + 0.591·19^-s − 0.581i·22^-s + 0.163i·23^-s + 0.920·26^-s + 0.350i·28^-s − 1.19·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1350$ = $2 \cdot 3^{3} \cdot 5^{2}$
Sign:	$0.447 - 0.894i$
Analytic conductor:	$79.6525$
Root analytic conductor:	$8.92482$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1350} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1350,\ (\ :3/2),\ 0.447 - 0.894i)$

Particular Values

$L(2)$	$\approx$	$0.9948106103$
$L(\frac12)$	$\approx$	$0.9948106103$
$L(\frac{5}{2})$		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 + 2iT $
	3	$ 1 $
	5	$ 1 $
good	7	$ 1 + 13iT - 343T^{2} $
	11	$ 1 - 30T + 1.33e3T^{2} $
	13	$ 1 - 61iT - 2.19e3T^{2} $
	17	$ 1 + 12iT - 4.91e3T^{2} $
	19	$ 1 - 49T + 6.85e3T^{2} $
	23	$ 1 - 18iT - 1.21e4T^{2} $
	29	$ 1 + 186T + 2.43e4T^{2} $
	31	$ 1 + 160T + 2.97e4T^{2} $
	37	$ 1 + 91iT - 5.06e4T^{2} $
	41	$ 1 + 378T + 6.89e4T^{2} $
	43	$ 1 - 268iT - 7.95e4T^{2} $
	47	$ 1 + 144iT - 1.03e5T^{2} $
	53	$ 1 - 570iT - 1.48e5T^{2} $
	59	$ 1 - 204T + 2.05e5T^{2} $
	61	$ 1 + 877T + 2.26e5T^{2} $
	67	$ 1 + 187iT - 3.00e5T^{2} $
	71	$ 1 - 606T + 3.57e5T^{2} $
	73	$ 1 + 431iT - 3.89e5T^{2} $
	79	$ 1 + 1.15e3T + 4.93e5T^{2} $
	83	$ 1 - 102iT - 5.71e5T^{2} $
	89	$ 1 - 984T + 7.04e5T^{2} $
	97	$ 1 + 265iT - 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

−9.278196941876454424326107901711, −9.004440073690009238713780197191, −7.69059077828809909688700807896, −7.03506615574261596947264000540, −6.07659504837577410072065228084, −4.93447108708793348394713127346, −4.05935219333665362592150762020, −3.41038932048985069281587079859, −2.00453666748180950143322196512, −1.15653420751520522632989679516, 0.24154588075604833752331488406, 1.65123767115518037018487535282, 3.07132678374584473819179441639, 3.94477925098019763418744864850, 5.23765791618369901198248961824, 5.66269033410536037170703666193, 6.63929730654231367033701296876, 7.45539383893865934755729516213, 8.283132753826826760982727541983, 8.967069346701478006011235648731

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.9948106103\)
\(L(\frac12)\)	\(\approx\)	\(0.9948106103\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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