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Properties

Label 1-31-31.10-r0-0-0
Degree $1$
Conductor $31$
Sign $-0.499 + 0.866i$
Analytic cond. $0.143963$
Root an. cond. $0.143963$
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  + (0.309 + 0.951i)2-s + (−0.978 + 0.207i)3-s + (−0.809 + 0.587i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.913 + 0.406i)7-s + (−0.809 − 0.587i)8-s + (0.913 − 0.406i)9-s + (−0.978 − 0.207i)10-s + (−0.104 − 0.994i)11-s + (0.669 − 0.743i)12-s + (0.669 + 0.743i)13-s + (−0.104 + 0.994i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.104 + 0.994i)17-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.978 + 0.207i)3-s + (−0.809 + 0.587i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.913 + 0.406i)7-s + (−0.809 − 0.587i)8-s + (0.913 − 0.406i)9-s + (−0.978 − 0.207i)10-s + (−0.104 − 0.994i)11-s + (0.669 − 0.743i)12-s + (0.669 + 0.743i)13-s + (−0.104 + 0.994i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.104 + 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(31\)
Sign: $-0.499 + 0.866i$
Analytic conductor: \(0.143963\)
Root analytic conductor: \(0.143963\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 31,\ (0:\ ),\ -0.499 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3110370237 + 0.5382880390i\)
\(L(\frac12)\) \(\approx\) \(0.3110370237 + 0.5382880390i\)
\(L(1)\) \(\approx\) \(0.6034514472 + 0.5344453494i\)
\(L(1)\) \(\approx\) \(0.6034514472 + 0.5344453494i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 + (-0.978 + 0.207i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.913 + 0.406i)T \)
11 \( 1 + (-0.104 - 0.994i)T \)
13 \( 1 + (0.669 + 0.743i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (0.669 - 0.743i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.913 - 0.406i)T \)
59 \( 1 + (-0.978 + 0.207i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.913 - 0.406i)T \)
73 \( 1 + (-0.104 - 0.994i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (-0.978 - 0.207i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
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

−36.24320029802769065796389121025, −35.6058569808683897985311616582, −33.779419814769545755870531510386, −32.71708610605082383893812963499, −31.17662863362418395813916760153, −30.213571781056725489155528923456, −28.93192945063543985914287011636, −27.87071150710215217353712949464, −27.264579035522765226885331341031, −24.60858878317887351630515720342, −23.46007769912824361314362507822, −22.6793634977704134296817678388, −20.95520891211973848839806950683, −20.15516949634351737809908176036, −18.34506544900951357549914188216, −17.37202372725927627512602871463, −15.615633056435682783925675261647, −13.60905162800547447422009118931, −12.27127261348917055068066935334, −11.41450205462944942836807675666, −9.95160054772889519555023459350, −7.8749552570529213699195399357, −5.38088687439200949180934465211, −4.298850718067870172115084159330, −1.32588585234507067359422873172, 3.9532594390772642425517817320, 5.60098766832234001236873175728, 6.87773670743363827984138677727, 8.52731738891497678888153118368, 10.82352262749934549132098129246, 12.00031661497892592790821252126, 13.96209449937104000167947936788, 15.30050854792731329600796505866, 16.34619759159998746027301860486, 17.82766498168819010446119747056, 18.69203619978809376093431825129, 21.45731525884525415567104504034, 22.17794764041459807961344269522, 23.62209553133215726271550153178, 24.20744914362598494459972157253, 26.1862412657695435868517349919, 27.09122899643455057716353748650, 28.25533759355966490337580506695, 30.1516804858797758646847114488, 31.102289562994732026957308724036, 32.68650886972354194059299581543, 33.83848465190598835468879863547, 34.565921281326361611227689466503, 35.33332501874698288304989885068, 37.16756181977935957341332185205

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