LMFDB - L-function 2-40e2-8.3-c2-0-24

Dirichlet series

L(s) = 1

− 1.90·3^-s + 3.95i·7^-s − 5.35·9^-s − 6.18·11^-s − 4.94i·13^-s − 22.4·17^-s + 10.3·19^-s − 7.54i·21^-s − 39.6i·23^-s + 27.4·27^-s + 30.4i·29^-s − 24.7i·31^-s + 11.8·33^-s + 24.0i·37^-s + 9.43i·39^-s + ⋯

L(s) = 1

− 0.636·3^-s + 0.565i·7^-s − 0.595·9^-s − 0.562·11^-s − 0.380i·13^-s − 1.31·17^-s + 0.546·19^-s − 0.359i·21^-s − 1.72i·23^-s + 1.01·27^-s + 1.04i·29^-s − 0.797i·31^-s + 0.357·33^-s + 0.649i·37^-s + 0.241i·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1600$ = $2^{6} \cdot 5^{2}$
Sign:	$0.965 - 0.258i$
Analytic conductor:	$43.5968$
Root analytic conductor:	$6.60279$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1600} (351, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1600,\ (\ :1),\ 0.965 - 0.258i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.9964431812$
$L(\frac12)$	$\approx$	$0.9964431812$
$L(2)$		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 $
	5	$ 1 $
good	3	$ 1 + 1.90T + 9T^{2} $
	7	$ 1 - 3.95iT - 49T^{2} $
	11	$ 1 + 6.18T + 121T^{2} $
	13	$ 1 + 4.94iT - 169T^{2} $
	17	$ 1 + 22.4T + 289T^{2} $
	19	$ 1 - 10.3T + 361T^{2} $
	23	$ 1 + 39.6iT - 529T^{2} $
	29	$ 1 - 30.4iT - 841T^{2} $
	31	$ 1 + 24.7iT - 961T^{2} $
	37	$ 1 - 24.0iT - 1.36e3T^{2} $
	41	$ 1 - 31.0T + 1.68e3T^{2} $
	43	$ 1 + 52.2T + 1.84e3T^{2} $
	47	$ 1 - 13.1iT - 2.20e3T^{2} $
	53	$ 1 - 17.9iT - 2.80e3T^{2} $
	59	$ 1 - 104.T + 3.48e3T^{2} $
	61	$ 1 - 57.2iT - 3.72e3T^{2} $
	67	$ 1 + 99.3T + 4.48e3T^{2} $
	71	$ 1 - 16.7iT - 5.04e3T^{2} $
	73	$ 1 + 96.3T + 5.32e3T^{2} $
	79	$ 1 + 139. iT - 6.24e3T^{2} $
	83	$ 1 - 91.7T + 6.88e3T^{2} $
	89	$ 1 - 94.7T + 7.92e3T^{2} $
	97	$ 1 - 143.T + 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

−9.009709707025302911427449561374, −8.649196972022588069613002268382, −7.67466611095297407331573921103, −6.64883998395258342095788033675, −5.99755192559546434509427834445, −5.18310852647358627903186844687, −4.47108789831693061277673699578, −3.04713171498589523599481614812, −2.27010810291942887761746113225, −0.58912588050623727882392802603, 0.52631721043164102357902738478, 1.97826990523149098615027337746, 3.18866847845211668458651605028, 4.23541504063291322482517468473, 5.16344031915732219392128026573, 5.85526418105898228824434300312, 6.78641698816236036654585985636, 7.47528050371874687415260379616, 8.405110828348122837288003854590, 9.213219812044311212245460561482

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.9964431812\)
\(L(\frac12)\)	\(\approx\)	\(0.9964431812\)
\(L(2)\)		not available
\(L(1)\)		not available

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