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Properties

Label 1-2720-2720.53-r1-0-0
Degree $1$
Conductor $2720$
Sign $0.392 + 0.919i$
Analytic cond. $292.304$
Root an. cond. $292.304$
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  + 3-s + (−0.707 − 0.707i)7-s + 9-s + i·11-s + (0.707 + 0.707i)13-s + (0.707 + 0.707i)19-s + (−0.707 − 0.707i)21-s + (0.707 + 0.707i)23-s + 27-s − i·29-s + (0.707 − 0.707i)31-s + i·33-s − i·37-s + (0.707 + 0.707i)39-s + (−0.707 + 0.707i)41-s + ⋯
L(s)  = 1  + 3-s + (−0.707 − 0.707i)7-s + 9-s + i·11-s + (0.707 + 0.707i)13-s + (0.707 + 0.707i)19-s + (−0.707 − 0.707i)21-s + (0.707 + 0.707i)23-s + 27-s − i·29-s + (0.707 − 0.707i)31-s + i·33-s − i·37-s + (0.707 + 0.707i)39-s + (−0.707 + 0.707i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2720\)    =    \(2^{5} \cdot 5 \cdot 17\)
Sign: $0.392 + 0.919i$
Analytic conductor: \(292.304\)
Root analytic conductor: \(292.304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2720} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2720,\ (1:\ ),\ 0.392 + 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.842579402 + 1.876704802i\)
\(L(\frac12)\) \(\approx\) \(2.842579402 + 1.876704802i\)
\(L(1)\) \(\approx\) \(1.533492933 + 0.1944792738i\)
\(L(1)\) \(\approx\) \(1.533492933 + 0.1944792738i\)

Euler product

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

−19.16350572478560832421734842925, −18.36727684960475169102865610475, −17.83538504521172240185088600712, −16.62213411803765856162310564779, −15.870188616283362697953376009875, −15.601186722777343930313474553591, −14.708823468608587962461209884418, −13.92714645317209632436609749241, −13.36521688868438161200807180145, −12.68633752493056077636911873735, −12.01777641809347849191739936792, −10.83337127367873674479358670849, −10.41233225386023318140405439827, −9.1422471832201076001338823567, −9.013003715516293236545354188167, −8.24323637746998709360559136382, −7.349783898587352064966452203071, −6.550079762062483621627773887, −5.74115097185483417971475758923, −4.901689542027849283515186847609, −3.75374667629365394814630221902, −3.04210811866930750593128743231, −2.69085403483341478679261247146, −1.39872495369795251619890487115, −0.50574033652792129720600982641, 1.004891939295448052790830970049, 1.73262278991572050910137889210, 2.74260557306958565404032846302, 3.56839259027308780869844744852, 4.14242346758805097279387788317, 4.969035435246366857064217579716, 6.29408840450775925447723430135, 6.851572332151367835249807939706, 7.66616393496581874416873045041, 8.21885123393938636594433243244, 9.36677933886087882616287824104, 9.67531035814964731857417960237, 10.32807150745795894561262350232, 11.39876743396630622357713859406, 12.18386245867761130169533837449, 13.140545960049717876393228689984, 13.49050894915185734349140566035, 14.17224750942352582802616413660, 15.02808524136225541755531845997, 15.58978630705709013914637696886, 16.33076905341163674821784022056, 17.042664142924880080199127247149, 17.90774783658283762656634423553, 18.78992562541848148831378226939, 19.18837541957902542045411130463

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