LMFDB - L-function 2-4410-21.20-c1-0-22

Dirichlet series

L(s) = 1

− i·2^-s − 4^-s − 5^-s + i·8^-s + i·10^-s − 0.930i·11^-s + 0.331i·13^-s + 16^-s − 3.26·17^-s − 0.547i·19^-s + 20^-s − 0.930·22^-s + 0.801i·23^-s + 25^-s + 0.331·26^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s − 0.447·5^-s + 0.353i·8^-s + 0.316i·10^-s − 0.280i·11^-s + 0.0920i·13^-s + 0.250·16^-s − 0.791·17^-s − 0.125i·19^-s + 0.223·20^-s − 0.198·22^-s + 0.167i·23^-s + 0.200·25^-s + 0.0650·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4410$ = $2 \cdot 3^{2} \cdot 5 \cdot 7^{2}$
Sign:	$0.442 + 0.896i$
Analytic conductor:	$35.2140$
Root analytic conductor:	$5.93414$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4410} (881, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4410,\ (\ :1/2),\ 0.442 + 0.896i)$

Particular Values

$L(1)$	$\approx$	$1.371106453$
$L(\frac12)$	$\approx$	$1.371106453$
$L(\frac{3}{2})$		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 + iT $
	3	$ 1 $
	5	$ 1 + T $
	7	$ 1 $
good	11	$ 1 + 0.930iT - 11T^{2} $
	13	$ 1 - 0.331iT - 13T^{2} $
	17	$ 1 + 3.26T + 17T^{2} $
	19	$ 1 + 0.547iT - 19T^{2} $
	23	$ 1 - 0.801iT - 23T^{2} $
	29	$ 1 - 6.37iT - 29T^{2} $
	31	$ 1 + 0.101iT - 31T^{2} $
	37	$ 1 - 8.29T + 37T^{2} $
	41	$ 1 - 6.57T + 41T^{2} $
	43	$ 1 - 1.49T + 43T^{2} $
	47	$ 1 + 6.57T + 47T^{2} $
	53	$ 1 + 6.38iT - 53T^{2} $
	59	$ 1 + 6.11T + 59T^{2} $
	61	$ 1 + 4.28iT - 61T^{2} $
	67	$ 1 + 2.25T + 67T^{2} $
	71	$ 1 - 3.64iT - 71T^{2} $
	73	$ 1 + 2.28iT - 73T^{2} $
	79	$ 1 - 4.32T + 79T^{2} $
	83	$ 1 - 15.4T + 83T^{2} $
	89	$ 1 - 0.219T + 89T^{2} $
	97	$ 1 - 3.65iT - 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

−8.304018245650629445153913543100, −7.65429288305924215613425645686, −6.78619987767964120451729350952, −6.01664238384763410694674979094, −5.03367757419152153377105730499, −4.39516691149330380349366588827, −3.55980942563844896575791985070, −2.79246739126506755356167618951, −1.79632316244459865337604768223, −0.60899451421699322310620580604, 0.70148823076699163221660999156, 2.15247235933625852422058288306, 3.18687774930502503176529875408, 4.29690034917271812928426354896, 4.58542163429624852256852236034, 5.73484539994263170710897919789, 6.29402056205799033958451443848, 7.08716014573061189028763716999, 7.78894808218647935779861964568, 8.253938675699537770533802426724

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.371106453\)
\(L(\frac12)\)	\(\approx\)	\(1.371106453\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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