LMFDB - L-function 2-4410-21.20-c1-0-5

Dirichlet series

L(s) = 1

− i·2^-s − 4^-s + 5^-s + i·8^-s − i·10^-s − 2.76i·11^-s − 6.49i·13^-s + 16^-s − 6.95·17^-s + 5.10i·19^-s − 20^-s − 2.76·22^-s + 9.08i·23^-s + 25^-s − 6.49·26^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s + 0.447·5^-s + 0.353i·8^-s − 0.316i·10^-s − 0.833i·11^-s − 1.80i·13^-s + 0.250·16^-s − 1.68·17^-s + 1.17i·19^-s − 0.223·20^-s − 0.589·22^-s + 1.89i·23^-s + 0.200·25^-s − 1.27·26^-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

$L(1)$	$\approx$	$0.7389158336$
$L(\frac12)$	$\approx$	$0.7389158336$
$L(\frac{3}{2})$		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 + 2.76iT - 11T^{2} $
	13	$ 1 + 6.49iT - 13T^{2} $
	17	$ 1 + 6.95T + 17T^{2} $
	19	$ 1 - 5.10iT - 19T^{2} $
	23	$ 1 - 9.08iT - 23T^{2} $
	29	$ 1 - 1.14iT - 29T^{2} $
	31	$ 1 - 9.75iT - 31T^{2} $
	37	$ 1 + 8.09T + 37T^{2} $
	41	$ 1 - 4.91T + 41T^{2} $
	43	$ 1 + 11.3T + 43T^{2} $
	47	$ 1 + 1.65T + 47T^{2} $
	53	$ 1 - 11.1iT - 53T^{2} $
	59	$ 1 - 9.50T + 59T^{2} $
	61	$ 1 + 8.28iT - 61T^{2} $
	67	$ 1 - 2.41T + 67T^{2} $
	71	$ 1 + 5.30iT - 71T^{2} $
	73	$ 1 + 8.34iT - 73T^{2} $
	79	$ 1 - 5.65T + 79T^{2} $
	83	$ 1 - 4.16T + 83T^{2} $
	89	$ 1 + 13.1T + 89T^{2} $
	97	$ 1 + 2.93iT - 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

−8.546653191423346504805517626177, −8.000406373758202265299405046819, −7.05877674064155057957479778932, −6.10897299883690665306959717461, −5.44780038178504582324208829442, −4.86826011752170103106344809245, −3.49873333742861614438247052524, −3.27838926836082285265247482113, −2.07302570817599021555179181959, −1.16432439832509647842642293220, 0.20745446011866191332804302881, 1.93844228380107709082720367427, 2.47659432891956989370145374039, 4.13383861568348027515560714004, 4.45481690616035278366358669523, 5.20757966330875247125995742503, 6.36904808340649342461421396543, 6.78959719127566735955306643513, 7.12291877416537215581742163171, 8.497525456736613841003706636100

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7389158336\)
\(L(\frac12)\)	\(\approx\)	\(0.7389158336\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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