LMFDB - L-function 2-1232-77.76-c1-0-20

Dirichlet series

L(s) = 1

− 0.646i·3^-s + 3.09i·5^-s + (2.44 + i)7^-s + 2.58·9^-s + (1.79 − 2.79i)11^-s − 0.646·13^-s + 2·15^-s + 3.74·17^-s − 1.80·19^-s + (0.646 − 1.58i)21^-s − 4·23^-s − 4.58·25^-s − 3.60i·27^-s − 1.58i·29^-s + 8.64i·31^-s + ⋯

L(s) = 1

− 0.373i·3^-s + 1.38i·5^-s + (0.925 + 0.377i)7^-s + 0.860·9^-s + (0.540 − 0.841i)11^-s − 0.179·13^-s + 0.516·15^-s + 0.907·17^-s − 0.413·19^-s + (0.140 − 0.345i)21^-s − 0.834·23^-s − 0.916·25^-s − 0.694i·27^-s − 0.293i·29^-s + 1.55i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.818 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1232$ = $2^{4} \cdot 7 \cdot 11$
Sign:	$0.818 - 0.575i$
Analytic conductor:	$9.83756$
Root analytic conductor:	$3.13649$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1232} (769, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1232,\ (\ :1/2),\ 0.818 - 0.575i)$

Particular Values

$L(1)$	$\approx$	$2.030456947$
$L(\frac12)$	$\approx$	$2.030456947$
$L(\frac{3}{2})$		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 $
	7	$ 1 + (-2.44 - i)T $
	11	$ 1 + (-1.79 + 2.79i)T $
good	3	$ 1 + 0.646iT - 3T^{2} $
	5	$ 1 - 3.09iT - 5T^{2} $
	13	$ 1 + 0.646T + 13T^{2} $
	17	$ 1 - 3.74T + 17T^{2} $
	19	$ 1 + 1.80T + 19T^{2} $
	23	$ 1 + 4T + 23T^{2} $
	29	$ 1 + 1.58iT - 29T^{2} $
	31	$ 1 - 8.64iT - 31T^{2} $
	37	$ 1 - 3.58T + 37T^{2} $
	41	$ 1 - 9.93T + 41T^{2} $
	43	$ 1 + 7.16iT - 43T^{2} $
	47	$ 1 - 9.93iT - 47T^{2} $
	53	$ 1 + 11.5T + 53T^{2} $
	59	$ 1 - 0.646iT - 59T^{2} $
	61	$ 1 + 1.93T + 61T^{2} $
	67	$ 1 - 7.58T + 67T^{2} $
	71	$ 1 + 2T + 71T^{2} $
	73	$ 1 + 16.1T + 73T^{2} $
	79	$ 1 + 4iT - 79T^{2} $
	83	$ 1 - 12.8T + 83T^{2} $
	89	$ 1 - 9.79iT - 89T^{2} $
	97	$ 1 + 8.50iT - 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

−9.942449646548361572082528447669, −8.941049640553494126292011376205, −7.912220594644896313245521593427, −7.43103250433402316372862289799, −6.46301156436344914307042261355, −5.86278584004332378291337088648, −4.58764777102887991460886866127, −3.54920005628262385003950632022, −2.51623765545792740860371514873, −1.36887718190932389726225457603, 1.05385169120322065533158172898, 1.97260308430280961287381442988, 3.93822658478200481481697297717, 4.46745365638027321483415470878, 5.09763320588151680224972696763, 6.18652881685002880080662350811, 7.50580344799846417028589602617, 7.903512140651843407664562775020, 8.937590111245412260106670698039, 9.647783971454582726300101122117

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.030456947\)
\(L(\frac12)\)	\(\approx\)	\(2.030456947\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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