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

Dirichlet series

L(s) = 1

− 32i·7^-s − 36·11^-s − 10i·13^-s − 78i·17^-s − 140·19^-s + 192i·23^-s + 6·29^-s − 16·31^-s + 34i·37^-s + 390·41^-s − 52i·43^-s + 408i·47^-s − 681·49^-s + 114i·53^-s + 516·59^-s + ⋯

L(s) = 1

− 1.72i·7^-s − 0.986·11^-s − 0.213i·13^-s − 1.11i·17^-s − 1.69·19^-s + 1.74i·23^-s + 0.0384·29^-s − 0.0926·31^-s + 0.151i·37^-s + 1.48·41^-s − 0.184i·43^-s + 1.26i·47^-s − 1.98·49^-s + 0.295i·53^-s + 1.13·59^-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

$L(2)$	$\approx$	$0.2031962914$
$L(\frac12)$	$\approx$	$0.2031962914$
$L(\frac{5}{2})$		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 + 32iT - 343T^{2} $
	11	$ 1 + 36T + 1.33e3T^{2} $
	13	$ 1 + 10iT - 2.19e3T^{2} $
	17	$ 1 + 78iT - 4.91e3T^{2} $
	19	$ 1 + 140T + 6.85e3T^{2} $
	23	$ 1 - 192iT - 1.21e4T^{2} $
	29	$ 1 - 6T + 2.43e4T^{2} $
	31	$ 1 + 16T + 2.97e4T^{2} $
	37	$ 1 - 34iT - 5.06e4T^{2} $
	41	$ 1 - 390T + 6.89e4T^{2} $
	43	$ 1 + 52iT - 7.95e4T^{2} $
	47	$ 1 - 408iT - 1.03e5T^{2} $
	53	$ 1 - 114iT - 1.48e5T^{2} $
	59	$ 1 - 516T + 2.05e5T^{2} $
	61	$ 1 + 58T + 2.26e5T^{2} $
	67	$ 1 - 892iT - 3.00e5T^{2} $
	71	$ 1 - 120T + 3.57e5T^{2} $
	73	$ 1 + 646iT - 3.89e5T^{2} $
	79	$ 1 - 1.16e3T + 4.93e5T^{2} $
	83	$ 1 - 732iT - 5.71e5T^{2} $
	89	$ 1 + 1.59e3T + 7.04e5T^{2} $
	97	$ 1 + 194iT - 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

−10.08251771920105367364786132381, −9.312911335494606885796893717926, −8.065837519507721514043263150276, −7.49617622850809430023169048554, −6.76668563484787632227539513324, −5.58943568063725422531029067487, −4.57250420162443293054703116360, −3.76858576405779076592406846087, −2.57100989214021799064565834971, −1.08117353147308754628981633304, 0.05536984929796623885427888700, 2.07520652165371850840818937811, 2.61306544119100572043465340250, 4.09808638382105583590281136722, 5.13593587030092903955385243444, 5.99306263793472195455416171496, 6.64685575370485945372785563117, 8.163608072462640338212797037514, 8.467651363494889881297109843323, 9.309957767920764281769896223534

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.2031962914\)
\(L(\frac12)\)	\(\approx\)	\(0.2031962914\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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