LMFDB - L-function 2-1428-17.16-c1-0-16

Dirichlet series

L(s) = 1

− i·3^-s + 2.60i·5^-s + i·7^-s − 9^-s − 5.34i·11^-s − 4.56·13^-s + 2.60·15^-s + (−3.06 + 2.75i)17^-s − 2.48·19^-s + 21^-s − 7.21i·23^-s − 1.78·25^-s + i·27^-s + 9.04i·29^-s − 9.03i·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 1.16i·5^-s + 0.377i·7^-s − 0.333·9^-s − 1.61i·11^-s − 1.26·13^-s + 0.672·15^-s + (−0.743 + 0.668i)17^-s − 0.569·19^-s + 0.218·21^-s − 1.50i·23^-s − 0.357·25^-s + 0.192i·27^-s + 1.68i·29^-s − 1.62i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1428 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 + 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1428 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1428$ = $2^{2} \cdot 3 \cdot 7 \cdot 17$
Sign:	$-0.743 + 0.668i$
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:	$(2,\ 1428,\ (\ :1/2),\ -0.743 + 0.668i)$

Particular Values

$L(1)$	$\approx$	$0.5680797029$
$L(\frac12)$	$\approx$	$0.5680797029$
$L(\frac{3}{2})$		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 + (3.06 - 2.75i)T $
good	5	$ 1 - 2.60iT - 5T^{2} $
	11	$ 1 + 5.34iT - 11T^{2} $
	13	$ 1 + 4.56T + 13T^{2} $
	19	$ 1 + 2.48T + 19T^{2} $
	23	$ 1 + 7.21iT - 23T^{2} $
	29	$ 1 - 9.04iT - 29T^{2} $
	31	$ 1 + 9.03iT - 31T^{2} $
	37	$ 1 + 5.69iT - 37T^{2} $
	41	$ 1 + 9.95iT - 41T^{2} $
	43	$ 1 - 4.29T + 43T^{2} $
	47	$ 1 + 9.03T + 47T^{2} $
	53	$ 1 - 5.78T + 53T^{2} $
	59	$ 1 + 10.8T + 59T^{2} $
	61	$ 1 + 14.3iT - 61T^{2} $
	67	$ 1 - 7.42T + 67T^{2} $
	71	$ 1 + 2.95iT - 71T^{2} $
	73	$ 1 - 7.17iT - 73T^{2} $
	79	$ 1 - 2.57iT - 79T^{2} $
	83	$ 1 + 5.55T + 83T^{2} $
	89	$ 1 + 17.1T + 89T^{2} $
	97	$ 1 - 3.24iT - 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

−9.037920717098089179515633610766, −8.436924005755131236620656937578, −7.53883482647160929722472658248, −6.67602611409608366929889699517, −6.19949178482903318958633088816, −5.23124218993865632642375485195, −3.89784076408369631708207520050, −2.82955938321202498875230224928, −2.18380348920733416775678467333, −0.21569982763079714681540040778, 1.55746281645278199443480717429, 2.77948500123564483275083260102, 4.34638577308590676074739700455, 4.62389332067703959559756021380, 5.37027882412882866999568183638, 6.70774145458472981010176305184, 7.47824952950984626067710540900, 8.295524115540676942372229573535, 9.329976422513223114977296486659, 9.676185474311316235528648385919

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.5680797029\)
\(L(\frac12)\)	\(\approx\)	\(0.5680797029\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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