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Properties

Label 1-231-231.5-r0-0-0
Degree $1$
Conductor $231$
Sign $0.717 + 0.696i$
Analytic cond. $1.07275$
Root an. cond. $1.07275$
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  + (−0.913 − 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.104 + 0.994i)5-s + (−0.309 − 0.951i)8-s + (0.5 − 0.866i)10-s + (0.809 − 0.587i)13-s + (−0.104 + 0.994i)16-s + (0.913 − 0.406i)17-s + (−0.669 + 0.743i)19-s + (−0.809 + 0.587i)20-s + (0.5 + 0.866i)23-s + (−0.978 − 0.207i)25-s + (−0.978 + 0.207i)26-s + (−0.309 + 0.951i)29-s + (0.104 + 0.994i)31-s + (0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.104 + 0.994i)5-s + (−0.309 − 0.951i)8-s + (0.5 − 0.866i)10-s + (0.809 − 0.587i)13-s + (−0.104 + 0.994i)16-s + (0.913 − 0.406i)17-s + (−0.669 + 0.743i)19-s + (−0.809 + 0.587i)20-s + (0.5 + 0.866i)23-s + (−0.978 − 0.207i)25-s + (−0.978 + 0.207i)26-s + (−0.309 + 0.951i)29-s + (0.104 + 0.994i)31-s + (0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.717 + 0.696i$
Analytic conductor: \(1.07275\)
Root analytic conductor: \(1.07275\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (0:\ ),\ 0.717 + 0.696i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7046277180 + 0.2855174515i\)
\(L(\frac12)\) \(\approx\) \(0.7046277180 + 0.2855174515i\)
\(L(1)\) \(\approx\) \(0.7273553924 + 0.08139132954i\)
\(L(1)\) \(\approx\) \(0.7273553924 + 0.08139132954i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.913 - 0.406i)T \)
5 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (-0.669 + 0.743i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 + (0.104 + 0.994i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (0.104 - 0.994i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.669 - 0.743i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
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

−26.0873123020646748136380371861, −25.48487515648702328774410221351, −24.33643312315769278498935822022, −23.869740307821017140529568689708, −22.796997594658227570514203776599, −21.0311177678807293050811943483, −20.72823692784243541438186156590, −19.38324078814568110577331587646, −18.86498475465955252322030913572, −17.53094776987294608614361605648, −16.848372480966082357307977194774, −16.03343953431726646125965956691, −15.136745621311972948072112876944, −13.92890120001762138515635508421, −12.71317391084327622064491727303, −11.59623268577704080424115115472, −10.578886697132493656476759320901, −9.35605598829263964870216930496, −8.65288181007227767367065491384, −7.72680649171861803589955473189, −6.43939190156594852394730775326, −5.42560797830740625667092242260, −4.098583707243524188386476192662, −2.165589090272450151658055464600, −0.830863544630072360048487668800, 1.41564770251563377688838310047, 2.90793959781575072288650053518, 3.696350676612471765559888440200, 5.75403385903061324056891944812, 6.93954355747862867351314979648, 7.80650897162909923285163690898, 8.8910399323924869684877919980, 10.147806793590703284978537475201, 10.758165896222197905436808500654, 11.74129442228963263986319735867, 12.802017202515513229475545886204, 14.124579880323120636349707785309, 15.25632742986474690311490711849, 16.144362495214930239527306921710, 17.26409402005197555201140780118, 18.21149717828679234153778275403, 18.82751650792650461378971573012, 19.71466732940575066678617603821, 20.82294316142913770062090299476, 21.5715313714771337961346274219, 22.70270375524708798466445096560, 23.54204229668309395257290054248, 25.16854265853127608178588197976, 25.55381935944991100708338110727, 26.55753133789535115068467745200

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