LMFDB - L-function 2-30e2-5.4-c3-0-15

Dirichlet series

L(s) = 1

− 2i·7^-s + 30·11^-s − 4i·13^-s − 90i·17^-s + 28·19^-s + 120i·23^-s − 210·29^-s − 4·31^-s − 200i·37^-s + 240·41^-s − 136i·43^-s + 120i·47^-s + 339·49^-s − 30i·53^-s + 450·59^-s + ⋯

L(s) = 1

− 0.107i·7^-s + 0.822·11^-s − 0.0853i·13^-s − 1.28i·17^-s + 0.338·19^-s + 1.08i·23^-s − 1.34·29^-s − 0.0231·31^-s − 0.888i·37^-s + 0.914·41^-s − 0.482i·43^-s + 0.372i·47^-s + 0.988·49^-s − 0.0777i·53^-s + 0.992·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$900$ = $2^{2} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$53.1017$
Root analytic conductor:	$7.28709$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{900} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 900,\ (\ :3/2),\ 0.447 + 0.894i)$

Particular Values

$L(2)$	$\approx$	$1.876204273$
$L(\frac12)$	$\approx$	$1.876204273$
$L(\frac{5}{2})$		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 $
	3	$ 1 $
	5	$ 1 $
good	7	$ 1 + 2iT - 343T^{2} $
	11	$ 1 - 30T + 1.33e3T^{2} $
	13	$ 1 + 4iT - 2.19e3T^{2} $
	17	$ 1 + 90iT - 4.91e3T^{2} $
	19	$ 1 - 28T + 6.85e3T^{2} $
	23	$ 1 - 120iT - 1.21e4T^{2} $
	29	$ 1 + 210T + 2.43e4T^{2} $
	31	$ 1 + 4T + 2.97e4T^{2} $
	37	$ 1 + 200iT - 5.06e4T^{2} $
	41	$ 1 - 240T + 6.89e4T^{2} $
	43	$ 1 + 136iT - 7.95e4T^{2} $
	47	$ 1 - 120iT - 1.03e5T^{2} $
	53	$ 1 + 30iT - 1.48e5T^{2} $
	59	$ 1 - 450T + 2.05e5T^{2} $
	61	$ 1 + 166T + 2.26e5T^{2} $
	67	$ 1 + 908iT - 3.00e5T^{2} $
	71	$ 1 + 1.02e3T + 3.57e5T^{2} $
	73	$ 1 + 250iT - 3.89e5T^{2} $
	79	$ 1 - 916T + 4.93e5T^{2} $
	83	$ 1 + 1.14e3iT - 5.71e5T^{2} $
	89	$ 1 - 420T + 7.04e5T^{2} $
	97	$ 1 + 1.53e3iT - 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

−9.327477117999820591859985928513, −9.077458509640400681824568308323, −7.64521413829382144285755038282, −7.21412527715072580097826691702, −6.05734782291819883945001197779, −5.22743037597371884508436182095, −4.11666844502765768121624199982, −3.18367518960115744679590450596, −1.85831384695908548445332944375, −0.55303527308519681293374018256, 1.08753953328986409167968713206, 2.28149160136605896831341064411, 3.61601024255644277325011327017, 4.40374737342277227871344860888, 5.63442383737230546666706575152, 6.39292657444229191763917115297, 7.29274695838758324888561995994, 8.303989356516527336828144194491, 8.984757123786402427642232785285, 9.862048838099089684281713303064

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.876204273\)
\(L(\frac12)\)	\(\approx\)	\(1.876204273\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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