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Properties

Label 1-2240-2240.109-r0-0-0
Degree $1$
Conductor $2240$
Sign $0.880 + 0.473i$
Analytic cond. $10.4025$
Root an. cond. $10.4025$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.991 + 0.130i)3-s + (0.965 − 0.258i)9-s + (0.608 − 0.793i)11-s + (0.923 + 0.382i)13-s + (−0.866 + 0.5i)17-s + (−0.793 + 0.608i)19-s + (0.965 − 0.258i)23-s + (−0.923 + 0.382i)27-s + (0.382 − 0.923i)29-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)33-s + (0.130 − 0.991i)37-s + (−0.965 − 0.258i)39-s + (0.707 + 0.707i)41-s + (0.382 + 0.923i)43-s + ⋯
L(s)  = 1  + (−0.991 + 0.130i)3-s + (0.965 − 0.258i)9-s + (0.608 − 0.793i)11-s + (0.923 + 0.382i)13-s + (−0.866 + 0.5i)17-s + (−0.793 + 0.608i)19-s + (0.965 − 0.258i)23-s + (−0.923 + 0.382i)27-s + (0.382 − 0.923i)29-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)33-s + (0.130 − 0.991i)37-s + (−0.965 − 0.258i)39-s + (0.707 + 0.707i)41-s + (0.382 + 0.923i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.880 + 0.473i$
Analytic conductor: \(10.4025\)
Root analytic conductor: \(10.4025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2240,\ (0:\ ),\ 0.880 + 0.473i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.169785620 + 0.2943059614i\)
\(L(\frac12)\) \(\approx\) \(1.169785620 + 0.2943059614i\)
\(L(1)\) \(\approx\) \(0.8693520076 + 0.06208113058i\)
\(L(1)\) \(\approx\) \(0.8693520076 + 0.06208113058i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.991 + 0.130i)T \)
11 \( 1 + (0.608 - 0.793i)T \)
13 \( 1 + (0.923 + 0.382i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.793 + 0.608i)T \)
23 \( 1 + (0.965 - 0.258i)T \)
29 \( 1 + (0.382 - 0.923i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.130 - 0.991i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (0.382 + 0.923i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.608 + 0.793i)T \)
59 \( 1 + (0.793 + 0.608i)T \)
61 \( 1 + (-0.608 - 0.793i)T \)
67 \( 1 + (-0.991 + 0.130i)T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (0.258 - 0.965i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (-0.923 - 0.382i)T \)
89 \( 1 + (0.258 + 0.965i)T \)
97 \( 1 + T \)
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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

−19.55628854030148253715131661167, −18.802153084850995832823065583079, −18.0775062829113671556251585808, −17.4064430403320825002674576495, −17.02013976087457773091986034932, −15.987973503736072611178759197567, −15.47258889934337835669376266769, −14.76067746441581216913440839495, −13.551373671039894928279861213908, −13.10806609596064525641533197779, −12.290084589671611231173284569486, −11.57011374144838236846972816922, −10.90723225249287226876062953843, −10.31542648975804500292820094160, −9.29620359412779519149725672296, −8.66964224804021894997379415342, −7.49273825642771356701896113592, −6.8005868629176381147818623212, −6.27282506879225674613813611548, −5.28463493017267884930991847841, −4.57172773420809391569594665793, −3.8589739755221316560023604991, −2.59116518480352631715936376566, −1.589440820072878378529289566359, −0.64961064746768687491421019305, 0.84799879788049149108664977361, 1.64091089373261265053293785979, 2.92894110439902228328852700553, 4.11234887491521894245745441672, 4.40049149588802444575357832531, 5.67102809793414185798651351079, 6.2595350928134330102082282133, 6.71490318899756279697886849611, 7.88052140527456506011926047338, 8.77904705242686510258990547317, 9.37896499121506287638399428354, 10.57103031104708569856512042115, 10.91731026162378774622686085593, 11.59354582265346784115027554402, 12.43964907792368216385711921411, 13.10332619802829751465776326863, 13.898060125793472997425914436447, 14.777129159167998494119591626074, 15.62588920500774779157344683517, 16.24435705274732131366947671806, 16.87576027104461049537943603340, 17.535444541078753036201869670844, 18.196768773496687710421922239799, 19.120786787397412701888944958188, 19.43086301023397906940437351644

Graph of the $Z$-function along the critical line