LMFDB - L-function 2-4400-5.4-c1-0-66

Dirichlet series

L(s) = 1

− 2.79i·3^-s + 0.208i·7^-s − 4.79·9^-s + 11^-s + i·13^-s + 0.791i·17^-s + 6.58·19^-s + 0.582·21^-s − 3.79i·23^-s + 4.99i·27^-s − 6.79·29^-s + 8.58·31^-s − 2.79i·33^-s − 2.58i·37^-s + 2.79·39^-s + ⋯

L(s) = 1

− 1.61i·3^-s + 0.0788i·7^-s − 1.59·9^-s + 0.301·11^-s + 0.277i·13^-s + 0.191i·17^-s + 1.51·19^-s + 0.127·21^-s − 0.790i·23^-s + 0.962i·27^-s − 1.26·29^-s + 1.54·31^-s − 0.485i·33^-s − 0.424i·37^-s + 0.446·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4400$ = $2^{4} \cdot 5^{2} \cdot 11$
Sign:	$-0.894 + 0.447i$
Analytic conductor:	$35.1341$
Root analytic conductor:	$5.92740$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4400} (4049, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4400,\ (\ :1/2),\ -0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$1.660758233$
$L(\frac12)$	$\approx$	$1.660758233$
$L(\frac{3}{2})$		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 $
	11	$ 1 - T $
good	3	$ 1 + 2.79iT - 3T^{2} $
	7	$ 1 - 0.208iT - 7T^{2} $
	13	$ 1 - iT - 13T^{2} $
	17	$ 1 - 0.791iT - 17T^{2} $
	19	$ 1 - 6.58T + 19T^{2} $
	23	$ 1 + 3.79iT - 23T^{2} $
	29	$ 1 + 6.79T + 29T^{2} $
	31	$ 1 - 8.58T + 31T^{2} $
	37	$ 1 + 2.58iT - 37T^{2} $
	41	$ 1 + 1.41T + 41T^{2} $
	43	$ 1 + 10iT - 43T^{2} $
	47	$ 1 - 1.41iT - 47T^{2} $
	53	$ 1 + 11.3iT - 53T^{2} $
	59	$ 1 + 10.5T + 59T^{2} $
	61	$ 1 - 4.20T + 61T^{2} $
	67	$ 1 - 4iT - 67T^{2} $
	71	$ 1 - 10.7T + 71T^{2} $
	73	$ 1 - 7.79iT - 73T^{2} $
	79	$ 1 + 15.5T + 79T^{2} $
	83	$ 1 + 9.95iT - 83T^{2} $
	89	$ 1 - 0.791T + 89T^{2} $
	97	$ 1 + 6.20iT - 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

−8.001691636017243026483317449072, −7.11252583285465168772017647942, −6.89415877206793976574350537666, −5.93818089134249958150696063022, −5.39339243518740575403218658570, −4.23830960898633452174474143019, −3.19380512017666953124710520274, −2.30370194219875638942630951753, −1.47798096105292030401073893233, −0.51120410345282507646669023921, 1.19396089585866482458430720296, 2.77394191210548824842656234159, 3.41001153887090481038786686783, 4.16248956289390071705329839527, 4.91397410411789914065325994907, 5.50734546682872376662024379502, 6.26812650862192244829221147442, 7.37395342545581043961001092118, 7.994762833236819241757640961384, 8.947076518578473608542777152790

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.660758233\)
\(L(\frac12)\)	\(\approx\)	\(1.660758233\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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