LMFDB - L-function 2-2240-40.29-c1-0-42

Dirichlet series

L(s) = 1

− 3.11·3^-s + (0.660 + 2.13i)5^-s − i·7^-s + 6.70·9^-s + 0.780i·11^-s + 3.09·13^-s + (−2.05 − 6.65i)15^-s − 7.31i·17^-s + 0.207i·19^-s + 3.11i·21^-s − 1.52i·23^-s + (−4.12 + 2.82i)25^-s − 11.5·27^-s + 3.60i·29^-s − 7.94·31^-s + ⋯

L(s) = 1

− 1.79·3^-s + (0.295 + 0.955i)5^-s − 0.377i·7^-s + 2.23·9^-s + 0.235i·11^-s + 0.857·13^-s + (−0.531 − 1.71i)15^-s − 1.77i·17^-s + 0.0476i·19^-s + 0.679i·21^-s − 0.318i·23^-s + (−0.825 + 0.564i)25^-s − 2.22·27^-s + 0.669i·29^-s − 1.42·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$0.532 + 0.846i$
Analytic conductor:	$17.8864$
Root analytic conductor:	$4.22924$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2240} (1569, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2240,\ (\ :1/2),\ 0.532 + 0.846i)$

Particular Values

$L(1)$	$\approx$	$0.7432224543$
$L(\frac12)$	$\approx$	$0.7432224543$
$L(\frac{3}{2})$		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 + (-0.660 - 2.13i)T $
	7	$ 1 + iT $
good	3	$ 1 + 3.11T + 3T^{2} $
	11	$ 1 - 0.780iT - 11T^{2} $
	13	$ 1 - 3.09T + 13T^{2} $
	17	$ 1 + 7.31iT - 17T^{2} $
	19	$ 1 - 0.207iT - 19T^{2} $
	23	$ 1 + 1.52iT - 23T^{2} $
	29	$ 1 - 3.60iT - 29T^{2} $
	31	$ 1 + 7.94T + 31T^{2} $
	37	$ 1 + 3.70T + 37T^{2} $
	41	$ 1 - 6.24T + 41T^{2} $
	43	$ 1 - 6.61T + 43T^{2} $
	47	$ 1 - 2.61iT - 47T^{2} $
	53	$ 1 + 8.69T + 53T^{2} $
	59	$ 1 + 14.6iT - 59T^{2} $
	61	$ 1 + 6.37iT - 61T^{2} $
	67	$ 1 + 7.21T + 67T^{2} $
	71	$ 1 - 14.6T + 71T^{2} $
	73	$ 1 - 0.502iT - 73T^{2} $
	79	$ 1 - 11.4T + 79T^{2} $
	83	$ 1 + 13.8T + 83T^{2} $
	89	$ 1 + 16.3T + 89T^{2} $
	97	$ 1 + 10.2iT - 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

−9.274603970975977353742747705081, −7.76582720716586372266296484197, −7.02463028649524246055835225660, −6.62066569760185620623967268258, −5.76543581297676517554734056904, −5.14693093864281462005095458839, −4.22754017993169835330132452004, −3.16379686667288419052371098843, −1.72183419669380381955548245884, −0.41375435168925578430830142060, 0.989770869300621474091884897020, 1.85580802064571461953259492688, 3.82219841115336296324073369829, 4.46269445594553343694518968693, 5.59355771098442891513155000172, 5.75910466658758174417001614656, 6.44476350991626595131468706780, 7.53676488375662272691654751310, 8.461146661322649454212815179646, 9.196652942696891394113588350304

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7432224543\)
\(L(\frac12)\)	\(\approx\)	\(0.7432224543\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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