LMFDB - L-function 2-2400-8.5-c1-0-34

Dirichlet series

L(s) = 1

− i·3^-s + 0.746·7^-s − 9^-s − 5.36i·11^-s − 2.92i·13^-s − 2.13·17^-s − 1.73i·19^-s − 0.746i·21^-s + 7.49·23^-s + i·27^-s + 6.74i·29^-s − 2.64·31^-s − 5.36·33^-s + 1.07i·37^-s − 2.92·39^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.282·7^-s − 0.333·9^-s − 1.61i·11^-s − 0.811i·13^-s − 0.517·17^-s − 0.397i·19^-s − 0.162i·21^-s + 1.56·23^-s + 0.192i·27^-s + 1.25i·29^-s − 0.475·31^-s − 0.933·33^-s + 0.176i·37^-s − 0.468·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2400$ = $2^{5} \cdot 3 \cdot 5^{2}$
Sign:	$-0.821 + 0.570i$
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:	$(2,\ 2400,\ (\ :1/2),\ -0.821 + 0.570i)$

Particular Values

$L(1)$	$\approx$	$1.277245739$
$L(\frac12)$	$\approx$	$1.277245739$
$L(\frac{3}{2})$		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 - 0.746T + 7T^{2} $
	11	$ 1 + 5.36iT - 11T^{2} $
	13	$ 1 + 2.92iT - 13T^{2} $
	17	$ 1 + 2.13T + 17T^{2} $
	19	$ 1 + 1.73iT - 19T^{2} $
	23	$ 1 - 7.49T + 23T^{2} $
	29	$ 1 - 6.74iT - 29T^{2} $
	31	$ 1 + 2.64T + 31T^{2} $
	37	$ 1 - 1.07iT - 37T^{2} $
	41	$ 1 + 11.2T + 41T^{2} $
	43	$ 1 + 7.44iT - 43T^{2} $
	47	$ 1 - 1.73T + 47T^{2} $
	53	$ 1 - 7.72iT - 53T^{2} $
	59	$ 1 + 6.85iT - 59T^{2} $
	61	$ 1 + 6.45iT - 61T^{2} $
	67	$ 1 + 7.44iT - 67T^{2} $
	71	$ 1 + 13.2T + 71T^{2} $
	73	$ 1 - 0.690T + 73T^{2} $
	79	$ 1 + 2.64T + 79T^{2} $
	83	$ 1 - 5.85iT - 83T^{2} $
	89	$ 1 - 7.59T + 89T^{2} $
	97	$ 1 - 14.1T + 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

−8.711636286456799186165110104288, −7.936839977845112141929854188284, −7.08214822960570812415987998762, −6.41017783279591127998206783787, −5.48208830856537647071090102939, −4.91448855116982814611490616747, −3.44634228508358987142668241070, −2.93345015887056617409291653498, −1.55567784678942424456703100997, −0.42869395937112494562553423246, 1.58548260360480044960548834394, 2.53707084663430266790204403500, 3.75930312267292580042435349698, 4.60397814764430294895629274875, 5.03066632981753228759468918816, 6.22680624812043041519145717589, 7.00695707461713642215786326793, 7.67080775747604085817971507534, 8.699418540630842438060477390463, 9.276160270199888237534456934818

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.277245739\)
\(L(\frac12)\)	\(\approx\)	\(1.277245739\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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