LMFDB - L-function 2-2800-5.4-c1-0-32

Dirichlet series

L(s) = 1

− 1.56i·3^-s − i·7^-s + 0.561·9^-s + 6.12·11^-s − 2i·13^-s + 1.56i·17^-s + 3.56·19^-s − 1.56·21^-s + 1.43i·23^-s − 5.56i·27^-s − 3.43·29^-s + 9.12·31^-s − 9.56i·33^-s + 8.80i·37^-s − 3.12·39^-s + ⋯

L(s) = 1

− 0.901i·3^-s − 0.377i·7^-s + 0.187·9^-s + 1.84·11^-s − 0.554i·13^-s + 0.378i·17^-s + 0.817·19^-s − 0.340·21^-s + 0.299i·23^-s − 1.07i·27^-s − 0.638·29^-s + 1.63·31^-s − 1.66i·33^-s + 1.44i·37^-s − 0.500·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2800$ = $2^{4} \cdot 5^{2} \cdot 7$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$22.3581$
Root analytic conductor:	$4.72843$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2800} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2800,\ (\ :1/2),\ 0.447 + 0.894i)$

Particular Values

$L(1)$	$\approx$	$2.307767200$
$L(\frac12)$	$\approx$	$2.307767200$
$L(\frac{3}{2})$		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 + iT $
good	3	$ 1 + 1.56iT - 3T^{2} $
	11	$ 1 - 6.12T + 11T^{2} $
	13	$ 1 + 2iT - 13T^{2} $
	17	$ 1 - 1.56iT - 17T^{2} $
	19	$ 1 - 3.56T + 19T^{2} $
	23	$ 1 - 1.43iT - 23T^{2} $
	29	$ 1 + 3.43T + 29T^{2} $
	31	$ 1 - 9.12T + 31T^{2} $
	37	$ 1 - 8.80iT - 37T^{2} $
	41	$ 1 + 2.43T + 41T^{2} $
	43	$ 1 - 6.56iT - 43T^{2} $
	47	$ 1 - 8.24iT - 47T^{2} $
	53	$ 1 - 1.12iT - 53T^{2} $
	59	$ 1 - 11.3T + 59T^{2} $
	61	$ 1 - 11.1T + 61T^{2} $
	67	$ 1 + 7.87iT - 67T^{2} $
	71	$ 1 + 1.68T + 71T^{2} $
	73	$ 1 + 6.43iT - 73T^{2} $
	79	$ 1 + 5.68T + 79T^{2} $
	83	$ 1 + 1.31iT - 83T^{2} $
	89	$ 1 + 9.80T + 89T^{2} $
	97	$ 1 - 6iT - 97T^{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

−8.442004490897878620438887965390, −7.910914727899275087312089858127, −7.01194564757859043766357645120, −6.57603444304151916800840967957, −5.84390331609604569277444602572, −4.65110496010331723723461068862, −3.88331654587880621317949068624, −2.92609117601338841310720152516, −1.54546765895884468657678162668, −1.00244876753282514339032836496, 1.11281497066095856328870657308, 2.31501067941884029106351264902, 3.66808647126897038840813217162, 4.02580068298830885842620489189, 4.96455660050966067516727673827, 5.75379142903300602547713940141, 6.78492745121218711230331013765, 7.18274115301482560539431984636, 8.541229205261290089063530543563, 8.971995925196835771291625163652

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.307767200\)
\(L(\frac12)\)	\(\approx\)	\(2.307767200\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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