LMFDB - L-function 2-2400-40.29-c1-0-22

Dirichlet series

L(s) = 1

− 3^-s − 1.97i·7^-s + 9^-s − 1.43i·11^-s + 0.241·13^-s + 7.38i·17^-s − 3.04i·19^-s + 1.97i·21^-s − 0.874i·23^-s − 27^-s − 9.07i·29^-s + 7.44·31^-s + 1.43i·33^-s + 8.81·37^-s − 0.241·39^-s + ⋯

L(s) = 1

− 0.577·3^-s − 0.747i·7^-s + 0.333·9^-s − 0.431i·11^-s + 0.0669·13^-s + 1.79i·17^-s − 0.697i·19^-s + 0.431i·21^-s − 0.182i·23^-s − 0.192·27^-s − 1.68i·29^-s + 1.33·31^-s + 0.249i·33^-s + 1.44·37^-s − 0.0386·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0439 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0439 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2400$ = $2^{5} \cdot 3 \cdot 5^{2}$
Sign:	$-0.0439 + 0.999i$
Analytic conductor:	$19.1640$
Root analytic conductor:	$4.37768$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2400} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2400,\ (\ :1/2),\ -0.0439 + 0.999i)$

Particular Values

$L(1)$	$\approx$	$1.127036417$
$L(\frac12)$	$\approx$	$1.127036417$
$L(\frac{3}{2})$		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 $
	3	$ 1 + T $
	5	$ 1 $
good	7	$ 1 + 1.97iT - 7T^{2} $
	11	$ 1 + 1.43iT - 11T^{2} $
	13	$ 1 - 0.241T + 13T^{2} $
	17	$ 1 - 7.38iT - 17T^{2} $
	19	$ 1 + 3.04iT - 19T^{2} $
	23	$ 1 + 0.874iT - 23T^{2} $
	29	$ 1 + 9.07iT - 29T^{2} $
	31	$ 1 - 7.44T + 31T^{2} $
	37	$ 1 - 8.81T + 37T^{2} $
	41	$ 1 + 1.91T + 41T^{2} $
	43	$ 1 + 11.2T + 43T^{2} $
	47	$ 1 + 3.34iT - 47T^{2} $
	53	$ 1 + 9.20T + 53T^{2} $
	59	$ 1 - 6.43iT - 59T^{2} $
	61	$ 1 + 4.57iT - 61T^{2} $
	67	$ 1 + 4.86T + 67T^{2} $
	71	$ 1 - 8.21T + 71T^{2} $
	73	$ 1 + 4.12iT - 73T^{2} $
	79	$ 1 + 13.6T + 79T^{2} $
	83	$ 1 + 12.3T + 83T^{2} $
	89	$ 1 - 8.08T + 89T^{2} $
	97	$ 1 + 10.6iT - 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

−8.554888669945471792914242456198, −8.088235823665779315566489687669, −7.14660343074412116800886755491, −6.29485339998182880432004804164, −5.87006910945288925158709977243, −4.61352027032201354549806626018, −4.11743861790954181531059423306, −3.01527193444095307886628693636, −1.66797901703562953114662776494, −0.46266448977912713216788464810, 1.17183793564535200941335976860, 2.44533133794470125304600315975, 3.36204936956507031839958251848, 4.68511050460338068854112992482, 5.12267279057942305763495311433, 6.05294299187826060496208939929, 6.78550274568974365276960918682, 7.55167763016772425210983099617, 8.404151786616211730898220040822, 9.282206920930802711559038129993

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.127036417\)
\(L(\frac12)\)	\(\approx\)	\(1.127036417\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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