LMFDB - L-function 2-1008-84.83-c2-0-8

Dirichlet series

L(s) = 1

+ 0.718·5^-s + (−6.52 − 2.54i)7^-s + 13.1·11^-s + 12.0i·13^-s − 17.0·17^-s + 6.05·19^-s − 10.8·23^-s − 24.4·25^-s + 8.20i·29^-s + 36.6·31^-s + (−4.68 − 1.82i)35^-s + 45.0·37^-s − 62.4·41^-s − 12.0i·43^-s + 87.3i·47^-s + ⋯

L(s) = 1

+ 0.143·5^-s + (−0.931 − 0.362i)7^-s + 1.19·11^-s + 0.926i·13^-s − 1.00·17^-s + 0.318·19^-s − 0.473·23^-s − 0.979·25^-s + 0.282i·29^-s + 1.18·31^-s + (−0.133 − 0.0521i)35^-s + 1.21·37^-s − 1.52·41^-s − 0.279i·43^-s + 1.85i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0602 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0602 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1008$ = $2^{4} \cdot 3^{2} \cdot 7$
Sign:	$-0.0602 - 0.998i$
Analytic conductor:	$27.4660$
Root analytic conductor:	$5.24080$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1008} (1007, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1008,\ (\ :1),\ -0.0602 - 0.998i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.193412017$
$L(\frac12)$	$\approx$	$1.193412017$
$L(2)$		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 $
	7	$ 1 + (6.52 + 2.54i)T $
good	5	$ 1 - 0.718T + 25T^{2} $
	11	$ 1 - 13.1T + 121T^{2} $
	13	$ 1 - 12.0iT - 169T^{2} $
	17	$ 1 + 17.0T + 289T^{2} $
	19	$ 1 - 6.05T + 361T^{2} $
	23	$ 1 + 10.8T + 529T^{2} $
	29	$ 1 - 8.20iT - 841T^{2} $
	31	$ 1 - 36.6T + 961T^{2} $
	37	$ 1 - 45.0T + 1.36e3T^{2} $
	41	$ 1 + 62.4T + 1.68e3T^{2} $
	43	$ 1 + 12.0iT - 1.84e3T^{2} $
	47	$ 1 - 87.3iT - 2.20e3T^{2} $
	53	$ 1 - 74.7iT - 2.80e3T^{2} $
	59	$ 1 - 22.0iT - 3.48e3T^{2} $
	61	$ 1 - 57.2iT - 3.72e3T^{2} $
	67	$ 1 - 47.6iT - 4.48e3T^{2} $
	71	$ 1 - 66.4T + 5.04e3T^{2} $
	73	$ 1 - 36.0iT - 5.32e3T^{2} $
	79	$ 1 + 46.6iT - 6.24e3T^{2} $
	83	$ 1 - 38.2iT - 6.88e3T^{2} $
	89	$ 1 + 126.T + 7.92e3T^{2} $
	97	$ 1 - 126. iT - 9.40e3T^{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

−9.730149643691221710679575945854, −9.367432759943267394405180607456, −8.460849512068737743824194129648, −7.28784434362134585398441224307, −6.53538195018820240154826917346, −5.99942080668444771911747401132, −4.47278345132311104825250922988, −3.87653284758406705428278982309, −2.61293180338520901050235730392, −1.26499495132178373861496765068, 0.39208063661879508175121503776, 2.00922741283746482402665931594, 3.20643519489097496404706922197, 4.09089398523618698410241689969, 5.31464322088012573970591437809, 6.30014885068401141324123132552, 6.75349878685798230498018293439, 8.012869223771203253675477538621, 8.759444987266900842415819831572, 9.728800813488524323116925645330

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 $\F_{q}$
Belyi maps

\(L(\frac{3}{2})\)	\(\approx\)	\(1.193412017\)
\(L(\frac12)\)	\(\approx\)	\(1.193412017\)
\(L(2)\)		not available
\(L(1)\)		not available

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