LMFDB - L-function 2-2240-5.4-c1-0-11

Dirichlet series

L(s) = 1

− 3.25i·3^-s + (1.49 + 1.66i)5^-s + i·7^-s − 7.57·9^-s + 2.88·11^-s + 6.94i·13^-s + (5.41 − 4.86i)15^-s + 0.549i·17^-s − 4.98·19^-s + 3.25·21^-s − 2.33i·23^-s + (−0.533 + 4.97i)25^-s + 14.8i·27^-s − 5.14·29^-s − 5.40·31^-s + ⋯

L(s) = 1

− 1.87i·3^-s + (0.668 + 0.743i)5^-s + 0.377i·7^-s − 2.52·9^-s + 0.870·11^-s + 1.92i·13^-s + (1.39 − 1.25i)15^-s + 0.133i·17^-s − 1.14·19^-s + 0.709·21^-s − 0.487i·23^-s + (−0.106 + 0.994i)25^-s + 2.86i·27^-s − 0.954·29^-s − 0.970·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$0.743 - 0.668i$
Analytic conductor:	$17.8864$
Root analytic conductor:	$4.22924$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2240} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2240,\ (\ :1/2),\ 0.743 - 0.668i)$

Particular Values

$L(1)$	$\approx$	$1.313371035$
$L(\frac12)$	$\approx$	$1.313371035$
$L(\frac{3}{2})$		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 + (-1.49 - 1.66i)T $
	7	$ 1 - iT $
good	3	$ 1 + 3.25iT - 3T^{2} $
	11	$ 1 - 2.88T + 11T^{2} $
	13	$ 1 - 6.94iT - 13T^{2} $
	17	$ 1 - 0.549iT - 17T^{2} $
	19	$ 1 + 4.98T + 19T^{2} $
	23	$ 1 + 2.33iT - 23T^{2} $
	29	$ 1 + 5.14T + 29T^{2} $
	31	$ 1 + 5.40T + 31T^{2} $
	37	$ 1 - 8.31iT - 37T^{2} $
	41	$ 1 + 2.87T + 41T^{2} $
	43	$ 1 - 7.75iT - 43T^{2} $
	47	$ 1 - 8.24iT - 47T^{2} $
	53	$ 1 + 1.81iT - 53T^{2} $
	59	$ 1 - 9.04T + 59T^{2} $
	61	$ 1 - 2.11T + 61T^{2} $
	67	$ 1 - 13.1iT - 67T^{2} $
	71	$ 1 + 3.28T + 71T^{2} $
	73	$ 1 + 2.18iT - 73T^{2} $
	79	$ 1 - 2.04T + 79T^{2} $
	83	$ 1 + 3.38iT - 83T^{2} $
	89	$ 1 - 3.24T + 89T^{2} $
	97	$ 1 - 5.22iT - 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

−8.975364897478225551006350079226, −8.329658422412572274369603818588, −7.29588317165944197100813628530, −6.63064141086741522611670340386, −6.43928597414847067484379786339, −5.60457596941308117476262977549, −4.20872055386010354021942921472, −2.92187667982185873330877949388, −1.96970008729663027468946642195, −1.55535859624896390200128746649, 0.42562627953575604113097304527, 2.21566519932586755694601591781, 3.57854830357646789195008587773, 3.92940730040087000405542344986, 5.09011015316082014941943007053, 5.43759688025765945762596603652, 6.23248099358959799930294678933, 7.60864445731455838934300155360, 8.620788093155006874222961787724, 9.000325175949536512759765906336

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.313371035\)
\(L(\frac12)\)	\(\approx\)	\(1.313371035\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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