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Properties

Label 1-239-239.163-r0-0-0
Degree $1$
Conductor $239$
Sign $-0.308 - 0.951i$
Analytic cond. $1.10991$
Root an. cond. $1.10991$
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  + (−0.850 − 0.526i)2-s + (0.0922 + 0.995i)3-s + (0.445 + 0.895i)4-s + (−0.850 + 0.526i)5-s + (0.445 − 0.895i)6-s + (−0.273 − 0.961i)7-s + (0.0922 − 0.995i)8-s + (−0.982 + 0.183i)9-s + 10-s + (0.445 − 0.895i)11-s + (−0.850 + 0.526i)12-s + (−0.602 + 0.798i)13-s + (−0.273 + 0.961i)14-s + (−0.602 − 0.798i)15-s + (−0.602 + 0.798i)16-s + (−0.273 − 0.961i)17-s + ⋯
L(s)  = 1  + (−0.850 − 0.526i)2-s + (0.0922 + 0.995i)3-s + (0.445 + 0.895i)4-s + (−0.850 + 0.526i)5-s + (0.445 − 0.895i)6-s + (−0.273 − 0.961i)7-s + (0.0922 − 0.995i)8-s + (−0.982 + 0.183i)9-s + 10-s + (0.445 − 0.895i)11-s + (−0.850 + 0.526i)12-s + (−0.602 + 0.798i)13-s + (−0.273 + 0.961i)14-s + (−0.602 − 0.798i)15-s + (−0.602 + 0.798i)16-s + (−0.273 − 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 239 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 239 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(239\)
Sign: $-0.308 - 0.951i$
Analytic conductor: \(1.10991\)
Root analytic conductor: \(1.10991\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{239} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 239,\ (0:\ ),\ -0.308 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1644248010 - 0.2262379681i\)
\(L(\frac12)\) \(\approx\) \(0.1644248010 - 0.2262379681i\)
\(L(1)\) \(\approx\) \(0.4898044097 + 0.02834338876i\)
\(L(1)\) \(\approx\) \(0.4898044097 + 0.02834338876i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad239 \( 1 \)
good2 \( 1 + (-0.850 - 0.526i)T \)
3 \( 1 + (0.0922 + 0.995i)T \)
5 \( 1 + (-0.850 + 0.526i)T \)
7 \( 1 + (-0.273 - 0.961i)T \)
11 \( 1 + (0.445 - 0.895i)T \)
13 \( 1 + (-0.602 + 0.798i)T \)
17 \( 1 + (-0.273 - 0.961i)T \)
19 \( 1 + (-0.850 + 0.526i)T \)
23 \( 1 + (-0.982 + 0.183i)T \)
29 \( 1 + (0.932 - 0.361i)T \)
31 \( 1 + (-0.273 - 0.961i)T \)
37 \( 1 + (0.739 - 0.673i)T \)
41 \( 1 + (0.739 - 0.673i)T \)
43 \( 1 + (-0.850 - 0.526i)T \)
47 \( 1 + (0.0922 - 0.995i)T \)
53 \( 1 + (-0.982 - 0.183i)T \)
59 \( 1 + (0.932 + 0.361i)T \)
61 \( 1 + (0.0922 + 0.995i)T \)
67 \( 1 + (0.0922 - 0.995i)T \)
71 \( 1 + (-0.273 - 0.961i)T \)
73 \( 1 + (-0.602 + 0.798i)T \)
79 \( 1 + (-0.602 + 0.798i)T \)
83 \( 1 + (0.0922 - 0.995i)T \)
89 \( 1 + (-0.602 - 0.798i)T \)
97 \( 1 + (-0.273 + 0.961i)T \)
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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

−26.32278934959084281207930904345, −25.28194830888087553885246247099, −24.906851021743239803425329940173, −23.87065570662092710832135163535, −23.282799809526765477894881861977, −22.07111075398424558942931879328, −20.30539229845373136320049298396, −19.656336116504058599377809992507, −19.13697113809288940518716309859, −17.93235320324778923483984715657, −17.39945743426347869194440542432, −16.19477326799336738320624730479, −15.1751896954831531304385254009, −14.585909618094176295266943033336, −12.796701581775399405485149234671, −12.271676408945592512110186518999, −11.192095956798660028319651209486, −9.742170353092782120378894561223, −8.57111978563090003807558868042, −8.09219676951369871759468696737, −6.93999232519729926423193389786, −6.054762030875409459242978116688, −4.75225113550115859462421003912, −2.70961626751746696706174986668, −1.481027053696309466090028668884, 0.26059578973152245751879728643, 2.51844455049423548337110556940, 3.73703670011639373071729760361, 4.25834570386961969846407397026, 6.40258858236927606805459802990, 7.512638058443320643904300662304, 8.5219548220106673213276200313, 9.60147136719321756650321617994, 10.44862252515979067070355001239, 11.292242103658595529187469909270, 11.96904197395559996593358332051, 13.68524794941207622205447122757, 14.656377466355221886525100125049, 15.99856795222264878962128670470, 16.44250710262608427130699343039, 17.34532049034391523516856349542, 18.7290333848386536369117940336, 19.579978847154605812461446637367, 20.104900232891531428234507774624, 21.20574019610250723711719925549, 22.059194341763619760018024561, 22.88438501352548825377052964399, 24.08371034155473471451950723208, 25.48338717025310250771757646237, 26.40631635839932496310725494292

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