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Properties

Label 1-4275-4275.1562-r0-0-0
Degree $1$
Conductor $4275$
Sign $0.954 - 0.299i$
Analytic cond. $19.8530$
Root an. cond. $19.8530$
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  + (−0.788 + 0.615i)2-s + (0.241 − 0.970i)4-s + i·7-s + (0.406 + 0.913i)8-s + (−0.669 + 0.743i)11-s + (0.139 − 0.990i)13-s + (−0.615 − 0.788i)14-s + (−0.882 − 0.469i)16-s + (−0.970 + 0.241i)17-s + (0.0697 − 0.997i)22-s + (−0.529 + 0.848i)23-s + (0.5 + 0.866i)26-s + (0.970 + 0.241i)28-s + (−0.719 − 0.694i)29-s + (0.913 − 0.406i)31-s + (0.984 − 0.173i)32-s + ⋯
L(s)  = 1  + (−0.788 + 0.615i)2-s + (0.241 − 0.970i)4-s + i·7-s + (0.406 + 0.913i)8-s + (−0.669 + 0.743i)11-s + (0.139 − 0.990i)13-s + (−0.615 − 0.788i)14-s + (−0.882 − 0.469i)16-s + (−0.970 + 0.241i)17-s + (0.0697 − 0.997i)22-s + (−0.529 + 0.848i)23-s + (0.5 + 0.866i)26-s + (0.970 + 0.241i)28-s + (−0.719 − 0.694i)29-s + (0.913 − 0.406i)31-s + (0.984 − 0.173i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4275\)    =    \(3^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.954 - 0.299i$
Analytic conductor: \(19.8530\)
Root analytic conductor: \(19.8530\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4275} (1562, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4275,\ (0:\ ),\ 0.954 - 0.299i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5972523096 - 0.09156841886i\)
\(L(\frac12)\) \(\approx\) \(0.5972523096 - 0.09156841886i\)
\(L(1)\) \(\approx\) \(0.5962127413 + 0.1830076904i\)
\(L(1)\) \(\approx\) \(0.5962127413 + 0.1830076904i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.788 + 0.615i)T \)
7 \( 1 + iT \)
11 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (0.139 - 0.990i)T \)
17 \( 1 + (-0.970 + 0.241i)T \)
23 \( 1 + (-0.529 + 0.848i)T \)
29 \( 1 + (-0.719 - 0.694i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (-0.848 + 0.529i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (-0.694 + 0.719i)T \)
53 \( 1 + (0.694 - 0.719i)T \)
59 \( 1 + (-0.882 - 0.469i)T \)
61 \( 1 + (0.0348 - 0.999i)T \)
67 \( 1 + (-0.829 + 0.559i)T \)
71 \( 1 + (-0.438 - 0.898i)T \)
73 \( 1 + (-0.788 + 0.615i)T \)
79 \( 1 + (-0.438 - 0.898i)T \)
83 \( 1 + (0.587 - 0.809i)T \)
89 \( 1 + (0.0348 - 0.999i)T \)
97 \( 1 + (0.829 + 0.559i)T \)
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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

−18.282235934607853006943033461506, −17.96218614206341559779673099086, −16.99033448969417608463127079365, −16.45460766455274212283684953797, −16.10015832286919417572047771905, −15.12567907637403080014213381240, −14.052436238478076515307422192069, −13.548673370436631501049283842716, −12.97331221497211371935763045198, −12.07629541907611190266241745782, −11.34729751778503321997586941306, −10.80008183267830064535068654038, −10.29039195542083860960181931772, −9.46260042491602764570076351762, −8.73569547017182100103820439981, −8.177587466543396439994814024622, −7.30213913359274909918412175777, −6.792034263242776732340377025095, −5.95150172621858230940287972343, −4.60384520015524395667125087941, −4.18349846389004346580910567965, −3.25411973103296981593674058050, −2.50183700334669425262414186636, −1.6279141082043414946435511112, −0.721560762945048494898894761405, 0.295403730414531292748699691406, 1.66063107541659665273732763921, 2.26961327499118894341297203846, 3.0888929774394851146782835844, 4.434686066281814345908734699972, 5.11566125444424358707327743242, 5.89851522407444290742954433714, 6.329985100162365448091921633418, 7.41200696769779927058107384767, 7.94511204196807293958471128413, 8.534834084890874617891092871855, 9.37909788741652451968854693396, 9.896589662941346799390613790213, 10.60698511291046368171360836239, 11.41693642912054281343732996568, 12.03086403737622419100999345930, 13.0994558644524858650499775912, 13.444556528361615299447471222745, 14.727526771039680943512125688791, 15.072217803424946413421179830272, 15.718025414203955396505371193587, 16.03993465592551681855103968093, 17.357987462078907175124595403377, 17.48474063170985228242780842470, 18.304011356598739295016798903456

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