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Properties

Label 1-232-232.197-r0-0-0
Degree $1$
Conductor $232$
Sign $0.0872 + 0.996i$
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  + (0.900 + 0.433i)3-s + (0.222 + 0.974i)5-s + (−0.900 − 0.433i)7-s + (0.623 + 0.781i)9-s + (−0.623 + 0.781i)11-s + (−0.623 + 0.781i)13-s + (−0.222 + 0.974i)15-s + 17-s + (0.900 − 0.433i)19-s + (−0.623 − 0.781i)21-s + (−0.222 + 0.974i)23-s + (−0.900 + 0.433i)25-s + (0.222 + 0.974i)27-s + (−0.222 − 0.974i)31-s + (−0.900 + 0.433i)33-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)3-s + (0.222 + 0.974i)5-s + (−0.900 − 0.433i)7-s + (0.623 + 0.781i)9-s + (−0.623 + 0.781i)11-s + (−0.623 + 0.781i)13-s + (−0.222 + 0.974i)15-s + 17-s + (0.900 − 0.433i)19-s + (−0.623 − 0.781i)21-s + (−0.222 + 0.974i)23-s + (−0.900 + 0.433i)25-s + (0.222 + 0.974i)27-s + (−0.222 − 0.974i)31-s + (−0.900 + 0.433i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0872 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0872 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.0872 + 0.996i$
Analytic conductor: \(1.07740\)
Root analytic conductor: \(1.07740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 232,\ (0:\ ),\ 0.0872 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.059217246 + 0.9705092605i\)
\(L(\frac12)\) \(\approx\) \(1.059217246 + 0.9705092605i\)
\(L(1)\) \(\approx\) \(1.179132832 + 0.4980922229i\)
\(L(1)\) \(\approx\) \(1.179132832 + 0.4980922229i\)

Euler product

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

−25.95013140503240846798598471199, −25.03361578708847722869186745246, −24.55468373335266034534639571937, −23.55646621028167967359340406085, −22.33880368237608824119970447342, −21.20284716644817012981347694263, −20.48473422658088150802537915933, −19.568367012672093268006767183159, −18.78883279434105976940153496742, −17.806903589559262562018101083341, −16.40940536820645468640321697143, −15.83220744644699618998263790575, −14.54824665718033779600606078397, −13.57790168755077328558695538527, −12.66664177303236295436731518678, −12.18902514087383511832945566600, −10.226865247221392378017653761328, −9.42329788306583577046661733669, −8.42529009055282275921170932184, −7.64272700795437280241394574135, −6.15442414657637398005754362411, −5.14467776992758456030729854554, −3.47389193900198323980907954380, −2.59464003747552330204889840913, −0.97379679224965357955743308576, 2.114618642992096387783987393064, 3.10165300667153457322900893582, 4.06755923329293433946476788870, 5.56413025392024507469558278791, 7.18910699682592698076006783118, 7.52022333152909041655929257999, 9.38768861577128765198651675970, 9.82976809952073475106055935878, 10.75977071033294705162112248770, 12.22653957836982188826335928043, 13.49969079709569541232984197802, 14.120971932083371472082014883295, 15.14275007068855270466581577709, 15.90996706344403970127176408902, 17.04559077360291055685314076680, 18.343307336856717013643142290241, 19.17462461393103923017770851570, 19.92247027967178094181845256022, 20.97992179520745773794592378322, 21.88254622616246445737868698359, 22.68458871196961268234607702377, 23.681737616963027753030109496579, 25.0124488034023226360110366967, 26.054494487405576173539388123310, 26.12527097645007549536912393479

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