LMFDB - L-function 2-40e2-5.4-c1-0-31

Dirichlet series

L(s) = 1

− 3i·3^-s + 2i·7^-s − 6·9^-s + 11^-s − 4i·13^-s − 5i·17^-s − 19^-s + 6·21^-s + 2i·23^-s + 9i·27^-s − 8·29^-s − 10·31^-s − 3i·33^-s − 6i·37^-s − 12·39^-s + ⋯

L(s) = 1

− 1.73i·3^-s + 0.755i·7^-s − 2·9^-s + 0.301·11^-s − 1.10i·13^-s − 1.21i·17^-s − 0.229·19^-s + 1.30·21^-s + 0.417i·23^-s + 1.73i·27^-s − 1.48·29^-s − 1.79·31^-s − 0.522i·33^-s − 0.986i·37^-s − 1.92·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1600$ = $2^{6} \cdot 5^{2}$
Sign:	$-0.894 - 0.447i$
Analytic conductor:	$12.7760$
Root analytic conductor:	$3.57436$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1600} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1600,\ (\ :1/2),\ -0.894 - 0.447i)$

Particular Values

$L(1)$	$\approx$	$0.8086102703$
$L(\frac12)$	$\approx$	$0.8086102703$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
good	3	$ 1 + 3iT - 3T^{2} $
	7	$ 1 - 2iT - 7T^{2} $
	11	$ 1 - T + 11T^{2} $
	13	$ 1 + 4iT - 13T^{2} $
	17	$ 1 + 5iT - 17T^{2} $
	19	$ 1 + T + 19T^{2} $
	23	$ 1 - 2iT - 23T^{2} $
	29	$ 1 + 8T + 29T^{2} $
	31	$ 1 + 10T + 31T^{2} $
	37	$ 1 + 6iT - 37T^{2} $
	41	$ 1 + 3T + 41T^{2} $
	43	$ 1 - 4iT - 43T^{2} $
	47	$ 1 - 4iT - 47T^{2} $
	53	$ 1 + 6iT - 53T^{2} $
	59	$ 1 + 8T + 59T^{2} $
	61	$ 1 + 10T + 61T^{2} $
	67	$ 1 - iT - 67T^{2} $
	71	$ 1 - 12T + 71T^{2} $
	73	$ 1 - 3iT - 73T^{2} $
	79	$ 1 - 6T + 79T^{2} $
	83	$ 1 + 13iT - 83T^{2} $
	89	$ 1 - 9T + 89T^{2} $
	97	$ 1 - 14iT - 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

−8.940365395135234342197874520385, −7.78267379702366503911566678537, −7.54441606307315142086291951611, −6.60787993037894861932008087359, −5.75553939452660216366465378292, −5.22950504693774832860943671498, −3.49425843478646385195998687414, −2.52736787156572640061655805974, −1.65215084025259195210162791554, −0.29693659920472739714115058474, 1.88065813762694887805252732370, 3.50182001433717918241826440946, 3.95092421656294872452447550579, 4.66676580480653568388087126255, 5.61698403181245981200100994309, 6.52023264273246910633162554055, 7.53401139642020225784072363159, 8.627158568023125130440847553623, 9.190256692326375913194274260451, 9.826948054714293204671499492915

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

\(L(1)\)	\(\approx\)	\(0.8086102703\)
\(L(\frac12)\)	\(\approx\)	\(0.8086102703\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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