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Properties

Label 1-232-232.181-r0-0-0
Degree $1$
Conductor $232$
Sign $0.999 + 0.00388i$
Analytic cond. $1.07740$
Root an. cond. $1.07740$
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  + (−0.623 − 0.781i)3-s + (0.900 − 0.433i)5-s + (0.623 + 0.781i)7-s + (−0.222 + 0.974i)9-s + (0.222 + 0.974i)11-s + (0.222 + 0.974i)13-s + (−0.900 − 0.433i)15-s + 17-s + (−0.623 + 0.781i)19-s + (0.222 − 0.974i)21-s + (−0.900 − 0.433i)23-s + (0.623 − 0.781i)25-s + (0.900 − 0.433i)27-s + (−0.900 + 0.433i)31-s + (0.623 − 0.781i)33-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)3-s + (0.900 − 0.433i)5-s + (0.623 + 0.781i)7-s + (−0.222 + 0.974i)9-s + (0.222 + 0.974i)11-s + (0.222 + 0.974i)13-s + (−0.900 − 0.433i)15-s + 17-s + (−0.623 + 0.781i)19-s + (0.222 − 0.974i)21-s + (−0.900 − 0.433i)23-s + (0.623 − 0.781i)25-s + (0.900 − 0.433i)27-s + (−0.900 + 0.433i)31-s + (0.623 − 0.781i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.999 + 0.00388i$
Analytic conductor: \(1.07740\)
Root analytic conductor: \(1.07740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 232,\ (0:\ ),\ 0.999 + 0.00388i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.207069754 + 0.002342002309i\)
\(L(\frac12)\) \(\approx\) \(1.207069754 + 0.002342002309i\)
\(L(1)\) \(\approx\) \(1.075835586 - 0.07916322911i\)
\(L(1)\) \(\approx\) \(1.075835586 - 0.07916322911i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 \)
good3 \( 1 + (-0.623 - 0.781i)T \)
5 \( 1 + (0.900 - 0.433i)T \)
7 \( 1 + (0.623 + 0.781i)T \)
11 \( 1 + (0.222 + 0.974i)T \)
13 \( 1 + (0.222 + 0.974i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.623 + 0.781i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + (-0.900 + 0.433i)T \)
37 \( 1 + (0.222 - 0.974i)T \)
41 \( 1 + T \)
43 \( 1 + (0.900 + 0.433i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (0.900 - 0.433i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.623 - 0.781i)T \)
67 \( 1 + (0.222 - 0.974i)T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (-0.900 - 0.433i)T \)
79 \( 1 + (-0.222 + 0.974i)T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−26.30496459180669166857230819290, −25.67711920213588084393029985926, −24.327503827281150223450463907429, −23.46173057547652543554607124874, −22.47291421982506454624537423623, −21.659002138477630832326465958245, −20.9899440978633347680152652545, −20.04195356092474117958836989725, −18.58555509575012914168508072139, −17.59415091444672444983159890786, −17.059083489899555257862414918538, −16.05603038419057877697186383524, −14.841289515648627847701540389996, −14.07899361957942885502439694169, −13.02001186903852499744288701235, −11.531793668572168981866647647968, −10.71872660630836481510296519939, −10.09323475842993770491251782355, −8.91709963262500942060064915478, −7.54974696299634346900638137427, −6.11792576788447836314801520450, −5.48923196900989483722322412363, −4.14891755000699347433956904307, −2.989727362331192066237017313417, −1.09202001212603957974614698793, 1.58738977048913143338711649246, 2.161228737099070583226071429939, 4.43228072032398574370313842111, 5.53989699581496309089939354036, 6.268902826740566532108102787070, 7.54024271079921578336645817973, 8.675781839827219573817413806378, 9.77008231359147573375994726619, 10.97674498468648614397427148561, 12.27633201781118244038753869917, 12.50664202838374572844290001201, 13.97138855707845295521931746196, 14.61783786061007503099698869978, 16.27205345646670372427938473377, 16.99029130512591978558278373089, 18.02029830827348323674204691317, 18.45564297174098121392684811030, 19.68311651854702567348181042444, 20.983447805673243623356475087243, 21.59166678879859034528267075816, 22.70223689584060822671781618911, 23.64281229994808332077233131294, 24.54668089571388070625522633201, 25.209302743642937107242360724380, 25.97007513777286025087684150136

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