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Properties

Label 1-183-183.131-r1-0-0
Degree $1$
Conductor $183$
Sign $-0.857 - 0.514i$
Analytic cond. $19.6660$
Root an. cond. $19.6660$
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  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.809 − 0.587i)5-s + (0.309 − 0.951i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)10-s − 11-s + 13-s − 14-s + (0.309 − 0.951i)16-s + (0.809 − 0.587i)17-s + (0.309 + 0.951i)19-s + (−0.309 + 0.951i)20-s + (0.309 + 0.951i)22-s + (0.809 − 0.587i)23-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.809 − 0.587i)5-s + (0.309 − 0.951i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)10-s − 11-s + 13-s − 14-s + (0.309 − 0.951i)16-s + (0.809 − 0.587i)17-s + (0.309 + 0.951i)19-s + (−0.309 + 0.951i)20-s + (0.309 + 0.951i)22-s + (0.809 − 0.587i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(183\)    =    \(3 \cdot 61\)
Sign: $-0.857 - 0.514i$
Analytic conductor: \(19.6660\)
Root analytic conductor: \(19.6660\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{183} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 183,\ (1:\ ),\ -0.857 - 0.514i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4230090125 - 1.526226959i\)
\(L(\frac12)\) \(\approx\) \(0.4230090125 - 1.526226959i\)
\(L(1)\) \(\approx\) \(0.7836168350 - 0.6816083520i\)
\(L(1)\) \(\approx\) \(0.7836168350 - 0.6816083520i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
61 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 - T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + (-0.809 - 0.587i)T \)
47 \( 1 - T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−27.309719608509907545441023007164, −25.97594566606287726701519482981, −25.773929861288001365866750143626, −24.72288225434721866472302873394, −23.731275145233503000547489040439, −22.805684422478304129665025180271, −21.71479646636292202472455594811, −20.99369662619791897489307684240, −19.22266998658690307114728034849, −18.344922345622987745142163252789, −17.89009853554324020432161536618, −16.71740710661632017962602143032, −15.501627874305020546775283616926, −14.93451197022196209407511505503, −13.72877503303832533094290728596, −12.96972050903828200781009112869, −11.19837236577000979512965582424, −10.1612265902735807377886697629, −9.096251564802077090113227801353, −8.13848539081887533181296450963, −6.89269355351348697641709474943, −5.775164373560573030381619845497, −5.12401956272438745998799489683, −3.12388707670626311429844916777, −1.51688156316794789288683849839, 0.672774079857476193867816681185, 1.7585466007942936563170669249, 3.256747175102639515816669101980, 4.56684667253873396872006821780, 5.67015513699841221293703534215, 7.54734649965992692474581733554, 8.504269011969623134086444765074, 9.74169838548733474492034115961, 10.45951743849495802612918760931, 11.493947432712081518262899017227, 12.89061154334092995242262630116, 13.42263560589729879956107194376, 14.40770205497275112170162807472, 16.34153586549684169535922281894, 16.9492867966222702408626388320, 18.125719343496621183970253880942, 18.70865149845208859994677815694, 20.30215310040451971164459591402, 20.72702615770893620310539391692, 21.361016244229297137934265216238, 22.81016151734228330007292137704, 23.479290139370830494247421949269, 24.856888620082093729233243871012, 25.93851059263110096709049562654, 26.68593069058823804205361715772

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