LMFDB - L-function 2-4004-13.12-c1-0-24

Dirichlet series

L(s) = 1

− 2.75·3^-s − 1.98i·5^-s − i·7^-s + 4.60·9^-s + i·11^-s + (−2.54 − 2.55i)13^-s + 5.46i·15^-s − 7.39·17^-s + 4.19i·19^-s + 2.75i·21^-s + 4.61·23^-s + 1.07·25^-s − 4.42·27^-s + 4.07·29^-s − 3.60i·31^-s + ⋯

L(s) = 1

− 1.59·3^-s − 0.886i·5^-s − 0.377i·7^-s + 1.53·9^-s + 0.301i·11^-s + (−0.704 − 0.709i)13^-s + 1.41i·15^-s − 1.79·17^-s + 0.963i·19^-s + 0.601i·21^-s + 0.963·23^-s + 0.214·25^-s − 0.851·27^-s + 0.755·29^-s − 0.647i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4004$ = $2^{2} \cdot 7 \cdot 11 \cdot 13$
Sign:	$0.704 + 0.709i$
Analytic conductor:	$31.9721$
Root analytic conductor:	$5.65438$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4004} (2157, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4004,\ (\ :1/2),\ 0.704 + 0.709i)$

Particular Values

$L(1)$	$\approx$	$0.7589341226$
$L(\frac12)$	$\approx$	$0.7589341226$
$L(\frac{3}{2})$		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 + iT $
	11	$ 1 - iT $
	13	$ 1 + (2.54 + 2.55i)T $
good	3	$ 1 + 2.75T + 3T^{2} $
	5	$ 1 + 1.98iT - 5T^{2} $
	17	$ 1 + 7.39T + 17T^{2} $
	19	$ 1 - 4.19iT - 19T^{2} $
	23	$ 1 - 4.61T + 23T^{2} $
	29	$ 1 - 4.07T + 29T^{2} $
	31	$ 1 + 3.60iT - 31T^{2} $
	37	$ 1 - 9.67iT - 37T^{2} $
	41	$ 1 - 5.43iT - 41T^{2} $
	43	$ 1 - 12.3T + 43T^{2} $
	47	$ 1 - 0.284iT - 47T^{2} $
	53	$ 1 + 4.46T + 53T^{2} $
	59	$ 1 - 1.96iT - 59T^{2} $
	61	$ 1 + 2.69T + 61T^{2} $
	67	$ 1 + 12.8iT - 67T^{2} $
	71	$ 1 - 8.00iT - 71T^{2} $
	73	$ 1 - 0.542iT - 73T^{2} $
	79	$ 1 + 13.5T + 79T^{2} $
	83	$ 1 + 3.32iT - 83T^{2} $
	89	$ 1 - 14.4iT - 89T^{2} $
	97	$ 1 - 15.3iT - 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

−8.307380595000448237163741983019, −7.50282257349179754923266273936, −6.68615625428160022587882227130, −6.15138317766376104052429261194, −5.25453907472986679043777793838, −4.70054215378723134728691104004, −4.25815694804899556504419280001, −2.76869990674786671753525445031, −1.41262363932963213911395364180, −0.52380652954286586927221298749, 0.58580232054965776565407735544, 2.12375416461109084408972161183, 2.93372325221350235851785838875, 4.33662154117921168679096796162, 4.78647300272984355470456241894, 5.68272654753363561829129196255, 6.36056056117958519326354831317, 7.02305187102446154595774810929, 7.24986541826764479471937054046, 8.846739205427400617503898138938

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7589341226\)
\(L(\frac12)\)	\(\approx\)	\(0.7589341226\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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