LMFDB - L-function 2-2240-5.4-c1-0-33

Dirichlet series

L(s) = 1

− 2.17i·3^-s + (1.45 − 1.69i)5^-s + i·7^-s − 1.74·9^-s + 2.13·11^-s + 5.52i·13^-s + (−3.68 − 3.17i)15^-s + 5.24i·17^-s + 6.50·19^-s + 2.17·21^-s + 6.35i·23^-s + (−0.741 − 4.94i)25^-s − 2.74i·27^-s + 9.09·29^-s + 6.77·31^-s + ⋯

L(s) = 1

− 1.25i·3^-s + (0.652 − 0.757i)5^-s + 0.377i·7^-s − 0.580·9^-s + 0.643·11^-s + 1.53i·13^-s + (−0.952 − 0.820i)15^-s + 1.27i·17^-s + 1.49·19^-s + 0.475·21^-s + 1.32i·23^-s + (−0.148 − 0.988i)25^-s − 0.527i·27^-s + 1.68·29^-s + 1.21·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$0.757 + 0.652i$
Analytic conductor:	$17.8864$
Root analytic conductor:	$4.22924$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2240} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2240,\ (\ :1/2),\ 0.757 + 0.652i)$

Particular Values

$L(1)$	$\approx$	$2.283801917$
$L(\frac12)$	$\approx$	$2.283801917$
$L(\frac{3}{2})$		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 + (-1.45 + 1.69i)T $
	7	$ 1 - iT $
good	3	$ 1 + 2.17iT - 3T^{2} $
	11	$ 1 - 2.13T + 11T^{2} $
	13	$ 1 - 5.52iT - 13T^{2} $
	17	$ 1 - 5.24iT - 17T^{2} $
	19	$ 1 - 6.50T + 19T^{2} $
	23	$ 1 - 6.35iT - 23T^{2} $
	29	$ 1 - 9.09T + 29T^{2} $
	31	$ 1 - 6.77T + 31T^{2} $
	37	$ 1 - 9.88iT - 37T^{2} $
	41	$ 1 + 2.35T + 41T^{2} $
	43	$ 1 + 0.354iT - 43T^{2} $
	47	$ 1 + 8.57iT - 47T^{2} $
	53	$ 1 - 7.37iT - 53T^{2} $
	59	$ 1 + 6.50T + 59T^{2} $
	61	$ 1 + 5.43T + 61T^{2} $
	67	$ 1 + 5.48iT - 67T^{2} $
	71	$ 1 + 3.11T + 71T^{2} $
	73	$ 1 - 7.37iT - 73T^{2} $
	79	$ 1 + 12.0T + 79T^{2} $
	83	$ 1 + 11.1iT - 83T^{2} $
	89	$ 1 + 6T + 89T^{2} $
	97	$ 1 + 0.979iT - 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

−8.795164654917350682022470934047, −8.255561507154860665724465879090, −7.33919039563482378634080884563, −6.46823937270062227298417579495, −6.13613204166272012504858140604, −5.05277837450673637548749967166, −4.17805968412591074983144013546, −2.84282438854785332686379965178, −1.53628408277633602534198719791, −1.36788012108938129339666527421, 0.932428178719764353349741626252, 2.80822870775679608000159198358, 3.16972018506483516739746548116, 4.34966706563497205142475638244, 5.06277546369397135033604829213, 5.84976116670999058180953992197, 6.77827596711098745762660873936, 7.51657090392818253233280532820, 8.504893597566441832147976808488, 9.485418735977752490581967454329

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.283801917\)
\(L(\frac12)\)	\(\approx\)	\(2.283801917\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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