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Properties

Label 1-861-861.11-r0-0-0
Degree $1$
Conductor $861$
Sign $0.0229 - 0.999i$
Analytic cond. $3.99846$
Root an. cond. $3.99846$
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  + (0.207 − 0.978i)2-s + (−0.913 − 0.406i)4-s + (−0.994 + 0.104i)5-s + (−0.587 + 0.809i)8-s + (−0.104 + 0.994i)10-s + (−0.777 + 0.629i)11-s + (−0.453 + 0.891i)13-s + (0.669 + 0.743i)16-s + (−0.629 − 0.777i)17-s + (0.998 + 0.0523i)19-s + (0.951 + 0.309i)20-s + (0.453 + 0.891i)22-s + (−0.978 − 0.207i)23-s + (0.978 − 0.207i)25-s + (0.777 + 0.629i)26-s + ⋯
L(s)  = 1  + (0.207 − 0.978i)2-s + (−0.913 − 0.406i)4-s + (−0.994 + 0.104i)5-s + (−0.587 + 0.809i)8-s + (−0.104 + 0.994i)10-s + (−0.777 + 0.629i)11-s + (−0.453 + 0.891i)13-s + (0.669 + 0.743i)16-s + (−0.629 − 0.777i)17-s + (0.998 + 0.0523i)19-s + (0.951 + 0.309i)20-s + (0.453 + 0.891i)22-s + (−0.978 − 0.207i)23-s + (0.978 − 0.207i)25-s + (0.777 + 0.629i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0229 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0229 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(861\)    =    \(3 \cdot 7 \cdot 41\)
Sign: $0.0229 - 0.999i$
Analytic conductor: \(3.99846\)
Root analytic conductor: \(3.99846\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{861} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 861,\ (0:\ ),\ 0.0229 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6176461125 - 0.6320129709i\)
\(L(\frac12)\) \(\approx\) \(0.6176461125 - 0.6320129709i\)
\(L(1)\) \(\approx\) \(0.7200519547 - 0.3756602062i\)
\(L(1)\) \(\approx\) \(0.7200519547 - 0.3756602062i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.207 - 0.978i)T \)
5 \( 1 + (-0.994 + 0.104i)T \)
11 \( 1 + (-0.777 + 0.629i)T \)
13 \( 1 + (-0.453 + 0.891i)T \)
17 \( 1 + (-0.629 - 0.777i)T \)
19 \( 1 + (0.998 + 0.0523i)T \)
23 \( 1 + (-0.978 - 0.207i)T \)
29 \( 1 + (0.987 + 0.156i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (0.951 - 0.309i)T \)
47 \( 1 + (0.838 - 0.544i)T \)
53 \( 1 + (0.358 + 0.933i)T \)
59 \( 1 + (-0.669 + 0.743i)T \)
61 \( 1 + (0.743 - 0.669i)T \)
67 \( 1 + (0.933 - 0.358i)T \)
71 \( 1 + (0.156 + 0.987i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.965 - 0.258i)T \)
83 \( 1 - T \)
89 \( 1 + (0.0523 - 0.998i)T \)
97 \( 1 + (0.156 - 0.987i)T \)
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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

−22.38203851430479254208974586542, −21.79691626358209111937949830500, −20.700452802776405962127235826886, −19.76184181925608446884016066030, −19.04355622788883701064437433269, −18.05113163599795662360049551153, −17.50245223727127756546825518112, −16.36931668341089745468765311553, −15.77420074537930088455211004211, −15.32161475792235232564861916089, −14.31905937708150837467617725882, −13.50035309730250946421614965506, −12.62674591295540093372421548929, −11.95573637275308385813299938207, −10.79337908526133067688337609785, −9.89312807521683188402225058754, −8.6456202037741005240953095503, −8.08266613687705796254299122876, −7.45094564915016797505765375207, −6.411246798337038007274984661453, −5.431162619931515846922357157204, −4.67248778859285963803533419881, −3.651986518461847649443952782282, −2.84082222119522936708440747765, −0.77259423097875873992797800629, 0.571077369223793958898493769618, 2.119064873721416423267269403595, 2.85104415185852147679124147942, 4.07425194374276834498226697697, 4.57707241252064571932985767715, 5.60174626560456033897508019536, 7.02243496664949497347836758854, 7.78899059091740532347584190053, 8.83805191873268132433979994338, 9.680206374508952637253078058568, 10.52303734880228886851987115846, 11.44182029753183231239218766422, 12.00717450294249664220910188620, 12.684955791600205990802396373813, 13.79264480135497049976156932442, 14.37591156462323652560356273164, 15.47869887109784809284034947164, 16.033920146496315346271116361065, 17.28168510635619686315823102169, 18.277076624844713235070273174209, 18.67820275611516084178516660098, 19.73550396728550536385599346028, 20.16380003954643079750246000996, 20.92527604462995520243353269514, 21.9106309843688186612093906337

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