LMFDB - L-function 2-2240-40.29-c1-0-36

Dirichlet series

L(s) = 1

− 1.29·3^-s + (0.437 + 2.19i)5^-s + i·7^-s − 1.32·9^-s + 0.711i·11^-s + 1.85·13^-s + (−0.564 − 2.83i)15^-s − 3.75i·17^-s − 7.57i·19^-s − 1.29i·21^-s − 8.45i·23^-s + (−4.61 + 1.91i)25^-s + 5.59·27^-s + 5.24i·29^-s − 0.349·31^-s + ⋯

L(s) = 1

− 0.746·3^-s + (0.195 + 0.980i)5^-s + 0.377i·7^-s − 0.443·9^-s + 0.214i·11^-s + 0.514·13^-s + (−0.145 − 0.731i)15^-s − 0.911i·17^-s − 1.73i·19^-s − 0.282i·21^-s − 1.76i·23^-s + (−0.923 + 0.383i)25^-s + 1.07·27^-s + 0.974i·29^-s − 0.0626·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0650i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$0.997 + 0.0650i$
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:	$(2,\ 2240,\ (\ :1/2),\ 0.997 + 0.0650i)$

Particular Values

$L(1)$	$\approx$	$1.199434816$
$L(\frac12)$	$\approx$	$1.199434816$
$L(\frac{3}{2})$		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 + (-0.437 - 2.19i)T $
	7	$ 1 - iT $
good	3	$ 1 + 1.29T + 3T^{2} $
	11	$ 1 - 0.711iT - 11T^{2} $
	13	$ 1 - 1.85T + 13T^{2} $
	17	$ 1 + 3.75iT - 17T^{2} $
	19	$ 1 + 7.57iT - 19T^{2} $
	23	$ 1 + 8.45iT - 23T^{2} $
	29	$ 1 - 5.24iT - 29T^{2} $
	31	$ 1 + 0.349T + 31T^{2} $
	37	$ 1 - 7.50T + 37T^{2} $
	41	$ 1 - 2.74T + 41T^{2} $
	43	$ 1 + 2.07T + 43T^{2} $
	47	$ 1 + 4.69iT - 47T^{2} $
	53	$ 1 - 9.81T + 53T^{2} $
	59	$ 1 - 6.54iT - 59T^{2} $
	61	$ 1 - 6.50iT - 61T^{2} $
	67	$ 1 - 10.4T + 67T^{2} $
	71	$ 1 + 5.53T + 71T^{2} $
	73	$ 1 - 9.10iT - 73T^{2} $
	79	$ 1 - 4.54T + 79T^{2} $
	83	$ 1 - 4.92T + 83T^{2} $
	89	$ 1 + 5.58T + 89T^{2} $
	97	$ 1 - 14.1iT - 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

−9.000835971673506659838417098440, −8.379262749221513055687964133628, −7.09683004100195923629106263573, −6.76398174269505100280335670358, −5.91963789265307993583710852368, −5.19185427615246855450800426694, −4.30583393548486448336999330433, −2.92548855870393747646249769971, −2.45602952880342400745863700308, −0.62683720822389215002129127671, 0.877706056618856700282567451022, 1.86968728579794760787251840888, 3.50275734361516166473184001206, 4.19744150625278063806372162822, 5.28468934769011410021540269851, 5.87413970417438435947836794335, 6.33911009823960679216248886730, 7.79884179730894603867449716912, 8.110196646287625683942655963656, 9.072300563787508858019262224884

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.199434816\)
\(L(\frac12)\)	\(\approx\)	\(1.199434816\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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