LMFDB - L-function 2-2400-8.5-c1-0-14

Dirichlet series

L(s) = 1

− i·3^-s + 2.64·7^-s − 9^-s − 1.51i·11^-s + 3.87i·13^-s − 3.31·17^-s + 7.08i·19^-s − 2.64i·21^-s + 4.82·23^-s + i·27^-s + 2.18i·29^-s + 7.36·31^-s − 1.51·33^-s + 7.87i·37^-s + 3.87·39^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.998·7^-s − 0.333·9^-s − 0.456i·11^-s + 1.07i·13^-s − 0.803·17^-s + 1.62i·19^-s − 0.576i·21^-s + 1.00·23^-s + 0.192i·27^-s + 0.405i·29^-s + 1.32·31^-s − 0.263·33^-s + 1.29i·37^-s + 0.619·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2400$ = $2^{5} \cdot 3 \cdot 5^{2}$
Sign:	$0.947 - 0.318i$
Analytic conductor:	$19.1640$
Root analytic conductor:	$4.37768$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2400} (1201, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2400,\ (\ :1/2),\ 0.947 - 0.318i)$

Particular Values

$L(1)$	$\approx$	$1.902801824$
$L(\frac12)$	$\approx$	$1.902801824$
$L(\frac{3}{2})$		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 + iT $
	5	$ 1 $
good	7	$ 1 - 2.64T + 7T^{2} $
	11	$ 1 + 1.51iT - 11T^{2} $
	13	$ 1 - 3.87iT - 13T^{2} $
	17	$ 1 + 3.31T + 17T^{2} $
	19	$ 1 - 7.08iT - 19T^{2} $
	23	$ 1 - 4.82T + 23T^{2} $
	29	$ 1 - 2.18iT - 29T^{2} $
	31	$ 1 - 7.36T + 31T^{2} $
	37	$ 1 - 7.87iT - 37T^{2} $
	41	$ 1 - 8.72T + 41T^{2} $
	43	$ 1 + 1.01iT - 43T^{2} $
	47	$ 1 + 7.08T + 47T^{2} $
	53	$ 1 - 4.50iT - 53T^{2} $
	59	$ 1 + 6.79iT - 59T^{2} $
	61	$ 1 - 3.60iT - 61T^{2} $
	67	$ 1 + 1.01iT - 67T^{2} $
	71	$ 1 - 6.72T + 71T^{2} $
	73	$ 1 + 15.5T + 73T^{2} $
	79	$ 1 - 7.36T + 79T^{2} $
	83	$ 1 + 7.74iT - 83T^{2} $
	89	$ 1 - 14.7T + 89T^{2} $
	97	$ 1 + 11.1T + 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

−8.781245414837879812338492920608, −8.242128916848808293446254050893, −7.55571170185809228113358492395, −6.63624551435543070223940123985, −6.05970749681238424863741398612, −4.98052902683378156122936493865, −4.31488604168282883974564583017, −3.15270736021931161335228520131, −2.00208579756048918977535494482, −1.19154009184101496486222041296, 0.72686286383059188983589445546, 2.25024580666397536498944080567, 3.07111662451272566507108576930, 4.38664057761789186623434900466, 4.79608898231414952497271702099, 5.59938462028951942243816235043, 6.64725777370482063194058009549, 7.47570414316772664255210045057, 8.231369885826383629012602900499, 8.934125195634839422994441899047

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.902801824\)
\(L(\frac12)\)	\(\approx\)	\(1.902801824\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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