LMFDB - L-function 2-2240-5.4-c1-0-70

Dirichlet series

L(s) = 1

− 2.44i·3^-s + (−2.22 − 0.224i)5^-s − i·7^-s − 2.99·9^-s − 4.89·11^-s − 4.44i·13^-s + (−0.550 + 5.44i)15^-s − 2i·17^-s − 1.55·19^-s − 2.44·21^-s − 2.89i·23^-s + (4.89 + i)25^-s + 6.89·29^-s − 8.89·31^-s + 11.9i·33^-s + ⋯

L(s) = 1

− 1.41i·3^-s + (−0.994 − 0.100i)5^-s − 0.377i·7^-s − 0.999·9^-s − 1.47·11^-s − 1.23i·13^-s + (−0.142 + 1.40i)15^-s − 0.485i·17^-s − 0.355·19^-s − 0.534·21^-s − 0.604i·23^-s + (0.979 + 0.200i)25^-s + 1.28·29^-s − 1.59·31^-s + 2.08i·33^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$0.100 - 0.994i$
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:	$(2,\ 2240,\ (\ :1/2),\ 0.100 - 0.994i)$

Particular Values

$L(1)$	$\approx$	$0.2342096602$
$L(\frac12)$	$\approx$	$0.2342096602$
$L(\frac{3}{2})$		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 + (2.22 + 0.224i)T $
	7	$ 1 + iT $
good	3	$ 1 + 2.44iT - 3T^{2} $
	11	$ 1 + 4.89T + 11T^{2} $
	13	$ 1 + 4.44iT - 13T^{2} $
	17	$ 1 + 2iT - 17T^{2} $
	19	$ 1 + 1.55T + 19T^{2} $
	23	$ 1 + 2.89iT - 23T^{2} $
	29	$ 1 - 6.89T + 29T^{2} $
	31	$ 1 + 8.89T + 31T^{2} $
	37	$ 1 - 2iT - 37T^{2} $
	41	$ 1 + 1.10T + 41T^{2} $
	43	$ 1 + 0.898iT - 43T^{2} $
	47	$ 1 - 8.89iT - 47T^{2} $
	53	$ 1 - 10.8iT - 53T^{2} $
	59	$ 1 - 1.55T + 59T^{2} $
	61	$ 1 + 3.55T + 61T^{2} $
	67	$ 1 - 8iT - 67T^{2} $
	71	$ 1 - 1.10T + 71T^{2} $
	73	$ 1 - 2.89iT - 73T^{2} $
	79	$ 1 - 6.89T + 79T^{2} $
	83	$ 1 + 2.44iT - 83T^{2} $
	89	$ 1 - 10T + 89T^{2} $
	97	$ 1 + 15.7iT - 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

−8.111159900199570790743225068152, −7.65126835033325308410922788456, −7.19534708049783395195133406631, −6.25154924433436304318965622659, −5.31861202553665876114168051871, −4.46304683750416693547648525364, −3.16540253905723505462947224391, −2.48961387616786359637251526014, −1.01771447463130669325798614860, −0.094002866957316846688170807967, 2.15483673423177973115574958474, 3.33740105094551033692078193691, 3.93740943058988220453295133434, 4.83168752410044702649472047185, 5.30904086575016875603804969980, 6.50804722989504676949278244825, 7.43467998397926474472214561327, 8.265493197091259019378217084591, 8.883170383829344345756038598822, 9.623715931130293094885658048634

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.2342096602\)
\(L(\frac12)\)	\(\approx\)	\(0.2342096602\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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