Properties

Label 1-231-231.83-r1-0-0
Degree $1$
Conductor $231$
Sign $0.957 - 0.288i$
Analytic cond. $24.8243$
Root an. cond. $24.8243$
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  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.809 − 0.587i)5-s + (0.309 + 0.951i)8-s + 10-s + (−0.809 + 0.587i)13-s + (−0.809 − 0.587i)16-s + (0.809 + 0.587i)17-s + (0.309 + 0.951i)19-s + (−0.809 + 0.587i)20-s − 23-s + (0.309 + 0.951i)25-s + (0.309 − 0.951i)26-s + (0.309 − 0.951i)29-s + (0.809 − 0.587i)31-s + 32-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.809 − 0.587i)5-s + (0.309 + 0.951i)8-s + 10-s + (−0.809 + 0.587i)13-s + (−0.809 − 0.587i)16-s + (0.809 + 0.587i)17-s + (0.309 + 0.951i)19-s + (−0.809 + 0.587i)20-s − 23-s + (0.309 + 0.951i)25-s + (0.309 − 0.951i)26-s + (0.309 − 0.951i)29-s + (0.809 − 0.587i)31-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.957 - 0.288i$
Analytic conductor: \(24.8243\)
Root analytic conductor: \(24.8243\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (1:\ ),\ 0.957 - 0.288i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8254082132 - 0.1214464053i\)
\(L(\frac12)\) \(\approx\) \(0.8254082132 - 0.1214464053i\)
\(L(1)\) \(\approx\) \(0.6349357117 + 0.06276918492i\)
\(L(1)\) \(\approx\) \(0.6349357117 + 0.06276918492i\)

Euler product

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

−26.39099673801793031613532260102, −25.49880885443033397184123057569, −24.44560854257441360040318327609, −23.24436736486954494057075961950, −22.24363136563980911020891859974, −21.51258158369392458903522909819, −20.04767254933799703174847989246, −19.810814291571356071730752577455, −18.58814124957106020757860898637, −17.99500969400985870277867398885, −16.81628202067758918268392231005, −15.87021304527594055670029624497, −14.93243693897893770956117801928, −13.602728486321286962519560344291, −12.2167041405830207659785548685, −11.710273560929815962018798311756, −10.533945965960857710683978891611, −9.79350581143331923280011551061, −8.4453397141945716316979554238, −7.5835185417127226757894981531, −6.72076454440428700906584096983, −4.83522698340335651094540639404, −3.41525892264444150653483510787, −2.5946615251947926900462856872, −0.803416004382760950763664799769, 0.53117506059949754847411866546, 1.97403199362596577270347927556, 3.90878833812662463755420531828, 5.14478633407913805897374942519, 6.3032419182764917679715423149, 7.65018802317303689931403411792, 8.15280806753953107208875104444, 9.39596781478308574531939739478, 10.24568002297838406044077514065, 11.60561644352125715317917993895, 12.34387940666494035784450972063, 13.95090647407317863072990801331, 14.877084478863085615827632269, 15.84875413947510100287355475748, 16.63843050861796481533234889870, 17.36779489038065681379775162838, 18.69568360403972862511592438436, 19.3264092544274333453043161841, 20.19529607694712494534951687889, 21.18168247771458184542004780659, 22.685050094131913424709004127526, 23.57598286833269579374942725031, 24.31986482709959164335803197572, 25.06531395878925342717584702401, 26.241740241817924846352645763122

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