LMFDB - L-function 1-4560-4560.2507-r0-0-0

Dirichlet series

L(s) = 1

+ i·7^-s + i·11^-s + 13^-s − i·17^-s + i·23^-s + i·29^-s + 31^-s + 37^-s − 41^-s − 43^-s + i·47^-s − 49^-s + 53^-s + i·59^-s − i·61^-s + ⋯

L(s) = 1

+ i·7^-s + i·11^-s + 13^-s − i·17^-s + i·23^-s + i·29^-s + 31^-s + 37^-s − 41^-s − 43^-s + i·47^-s − 49^-s + 53^-s + i·59^-s − i·61^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$1$
Conductor:	$4560$ = $2^{4} \cdot 3 \cdot 5 \cdot 19$
Sign:	$-0.160 + 0.987i$
Analytic conductor:	$21.1765$
Root analytic conductor:	$21.1765$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4560} (2507, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 4560,\ (0:\ ),\ -0.160 + 0.987i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.112989429 + 1.308162272i$
$L(\frac12)$	$\approx$	$1.112989429 + 1.308162272i$
$L(1)$	$\approx$	$1.077254640 + 0.3123913518i$
$L(1)$	$\approx$	$1.077254640 + 0.3123913518i$

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	5	$ 1 $
	19	$ 1 $
good	7	$ 1 $
	11	$ 1 $
	13	$ 1 $
	17	$ 1 $
	23	$ 1 $
	29	$ 1 $
	31	$ 1 $
	37	$ 1 + iT $
	41	$ 1 $
	43	$ 1 + T $
	47	$ 1 $
	53	$ 1 $
	59	$ 1 $
	61	$ 1 - iT $
	67	$ 1 $
	71	$ 1 $
	73	$ 1 $
	79	$ 1 $
	83	$ 1 $
	89	$ 1 + iT $
	97	$ 1 $
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$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−18.149438319840815320394688931088, −17.10804963237449319901551367766, −16.84090916387961051441394189015, −16.127411856510230069030772008289, −15.38912016269983857050522430397, −14.64632175391846569217322193604, −13.86137412550796691705713492779, −13.38684032825334451180246897440, −12.8727115628829122656406315160, −11.74507918645701860135035402322, −11.2859936203953341999763521275, −10.38152212163866378200202085114, −10.18840824492192005112058537042, −9.00894674903147558643000536973, −8.24855261665956940128128953160, −7.97790164433982747415272017292, −6.71445733039475172192542284081, −6.38217456280450853458404117795, −5.566732053334468157357599688342, −4.55503429587307395214725226485, −3.863559296240766174818899146022, −3.32628433112560525220156531813, −2.27254910895099117781951006072, −1.24177455750950287011028625856, −0.526634747869584277777036967616, 1.106341045992652478865908317631, 1.90779797317271665615892583515, 2.760571462654713401496944411062, 3.45468127164459549144801998205, 4.47857101337769734150390285867, 5.14562185800274939114978139553, 5.830184278187405497643250760304, 6.65363972706722116798687286891, 7.30184889539620878774683954167, 8.21655690468457006341772751588, 8.79391698346841867364502929506, 9.59027766814951932532815201936, 10.02756795431337662169931183015, 11.16249131021896339597820484397, 11.60018950553484420102935962075, 12.309261488308985793323043573968, 12.98126389535009332511708993542, 13.6780625707873046299669585312, 14.41518700870089984923832603283, 15.24631198544576398405426674969, 15.61654958889472877016444699537, 16.26912536376874235663789233464, 17.141323856974561880472866218913, 17.91798051990883159670992617193, 18.367032280233056086599640756197

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(1.112989429 + 1.308162272i\)
\(L(\frac12)\)	\(\approx\)	\(1.112989429 + 1.308162272i\)
\(L(1)\)	\(\approx\)	\(1.077254640 + 0.3123913518i\)
\(L(1)\)	\(\approx\)	\(1.077254640 + 0.3123913518i\)

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