LMFDB - L-function 2-1428-17.16-c1-0-9

Dirichlet series

L(s) = 1

− i·3^-s − 0.655i·5^-s + i·7^-s − 9^-s + 2.37i·11^-s + 4.29·13^-s − 0.655·15^-s + (3.97 + 1.09i)17^-s − 4.64·19^-s + 21^-s − 4.92i·23^-s + 4.57·25^-s + i·27^-s + 2.35i·29^-s + 9.33i·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 0.292i·5^-s + 0.377i·7^-s − 0.333·9^-s + 0.717i·11^-s + 1.19·13^-s − 0.169·15^-s + (0.964 + 0.265i)17^-s − 1.06·19^-s + 0.218·21^-s − 1.02i·23^-s + 0.914·25^-s + 0.192i·27^-s + 0.436i·29^-s + 1.67i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1428 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1428 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1428$ = $2^{2} \cdot 3 \cdot 7 \cdot 17$
Sign:	$0.964 + 0.265i$
Analytic conductor:	$11.4026$
Root analytic conductor:	$3.37677$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1428} (169, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1428,\ (\ :1/2),\ 0.964 + 0.265i)$

Particular Values

$L(1)$	$\approx$	$1.761715662$
$L(\frac12)$	$\approx$	$1.761715662$
$L(\frac{3}{2})$		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 $
	7	$ 1 - iT $
	17	$ 1 + (-3.97 - 1.09i)T $
good	5	$ 1 + 0.655iT - 5T^{2} $
	11	$ 1 - 2.37iT - 11T^{2} $
	13	$ 1 - 4.29T + 13T^{2} $
	19	$ 1 + 4.64T + 19T^{2} $
	23	$ 1 + 4.92iT - 23T^{2} $
	29	$ 1 - 2.35iT - 29T^{2} $
	31	$ 1 - 9.33iT - 31T^{2} $
	37	$ 1 - 0.725iT - 37T^{2} $
	41	$ 1 + 3.19iT - 41T^{2} $
	43	$ 1 - 9.56T + 43T^{2} $
	47	$ 1 - 9.33T + 47T^{2} $
	53	$ 1 - 3.27T + 53T^{2} $
	59	$ 1 + 7.26T + 59T^{2} $
	61	$ 1 + 7.45iT - 61T^{2} $
	67	$ 1 - 15.1T + 67T^{2} $
	71	$ 1 + 15.9iT - 71T^{2} $
	73	$ 1 + 4.94iT - 73T^{2} $
	79	$ 1 - 6.58iT - 79T^{2} $
	83	$ 1 + 7.82T + 83T^{2} $
	89	$ 1 - 3.57T + 89T^{2} $
	97	$ 1 - 2.32iT - 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

−9.253567075272696591244910989899, −8.662411501093000656947701398583, −8.020542085391013364424435509258, −6.97831568349829802593638924861, −6.31930278578061428813790750167, −5.43267436013113711210370485267, −4.47047144326525390622720869807, −3.35681376472900797314190006738, −2.17987641989268137103887987052, −1.05892533060773666056892946374, 0.950186226130312162552385409741, 2.61720083987826935963175589663, 3.66105242344247371559596761710, 4.26096381910003950942786776341, 5.61423035172375293505676805362, 6.07078184047538519993317821602, 7.19228750108335956251362367306, 8.064529776641575743667858998028, 8.800295306526275294285131336212, 9.608236112413522819563084745368

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.761715662\)
\(L(\frac12)\)	\(\approx\)	\(1.761715662\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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