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Properties

Label 1-235-235.119-r0-0-0
Degree $1$
Conductor $235$
Sign $-0.241 + 0.970i$
Analytic cond. $1.09133$
Root an. cond. $1.09133$
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  + (−0.203 + 0.979i)2-s + (−0.682 + 0.730i)3-s + (−0.917 − 0.398i)4-s + (−0.576 − 0.816i)6-s + (0.775 − 0.631i)7-s + (0.576 − 0.816i)8-s + (−0.0682 − 0.997i)9-s + (−0.334 + 0.942i)11-s + (0.917 − 0.398i)12-s + (0.990 − 0.136i)13-s + (0.460 + 0.887i)14-s + (0.682 + 0.730i)16-s + (0.334 + 0.942i)17-s + (0.990 + 0.136i)18-s + (0.962 − 0.269i)19-s + ⋯
L(s)  = 1  + (−0.203 + 0.979i)2-s + (−0.682 + 0.730i)3-s + (−0.917 − 0.398i)4-s + (−0.576 − 0.816i)6-s + (0.775 − 0.631i)7-s + (0.576 − 0.816i)8-s + (−0.0682 − 0.997i)9-s + (−0.334 + 0.942i)11-s + (0.917 − 0.398i)12-s + (0.990 − 0.136i)13-s + (0.460 + 0.887i)14-s + (0.682 + 0.730i)16-s + (0.334 + 0.942i)17-s + (0.990 + 0.136i)18-s + (0.962 − 0.269i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $-0.241 + 0.970i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ -0.241 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5329165649 + 0.6815407195i\)
\(L(\frac12)\) \(\approx\) \(0.5329165649 + 0.6815407195i\)
\(L(1)\) \(\approx\) \(0.6582654109 + 0.4807602216i\)
\(L(1)\) \(\approx\) \(0.6582654109 + 0.4807602216i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.203 + 0.979i)T \)
3 \( 1 + (-0.682 + 0.730i)T \)
7 \( 1 + (0.775 - 0.631i)T \)
11 \( 1 + (-0.334 + 0.942i)T \)
13 \( 1 + (0.990 - 0.136i)T \)
17 \( 1 + (0.334 + 0.942i)T \)
19 \( 1 + (0.962 - 0.269i)T \)
23 \( 1 + (-0.203 - 0.979i)T \)
29 \( 1 + (-0.990 - 0.136i)T \)
31 \( 1 + (0.682 + 0.730i)T \)
37 \( 1 + (-0.460 + 0.887i)T \)
41 \( 1 + (-0.576 - 0.816i)T \)
43 \( 1 + (0.917 + 0.398i)T \)
53 \( 1 + (0.576 + 0.816i)T \)
59 \( 1 + (-0.917 + 0.398i)T \)
61 \( 1 + (0.460 + 0.887i)T \)
67 \( 1 + (0.775 + 0.631i)T \)
71 \( 1 + (0.203 + 0.979i)T \)
73 \( 1 + (0.0682 - 0.997i)T \)
79 \( 1 + (0.854 - 0.519i)T \)
83 \( 1 + (0.334 - 0.942i)T \)
89 \( 1 + (0.962 + 0.269i)T \)
97 \( 1 + (-0.682 + 0.730i)T \)
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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

−26.16201272321847411195725714198, −24.937487616932528972437655693199, −24.03806353773720482050913498225, −23.11025774178044575098231687395, −22.22849900793600959767006177071, −21.31411685885153181089511165638, −20.52873329767510133112171968084, −19.209821511484335870071763535815, −18.4300264588675937502087446024, −18.04055405204588427375138715321, −16.87571438128776173995691678108, −15.83716300305928371094347158312, −14.039960538199434628179217319200, −13.50208401775303952227408785448, −12.311049573653498166269585728949, −11.40061123373153048919252636151, −11.06445763805094166812459867786, −9.55811408379925458259276194653, −8.36899908612842191658597313311, −7.57859830505307780977367869378, −5.803698745524438506664504426604, −5.10213026450593913479398675842, −3.45411890715605850146615465037, −2.10390800118008561106816387743, −0.967753696780810111031123773632, 1.19307133385337355614861135499, 3.79325343321368691006705692938, 4.6857551876770223019936688344, 5.60756571735329148131958417238, 6.73125450759366241026368906293, 7.83391214481166804756762396243, 8.91736886279318837072043975214, 10.16052611854566224837753686290, 10.73205302097541939414683416273, 12.13815101751955595135597261368, 13.42232251796899371407633660398, 14.53623521737679685231801847222, 15.32882687185495074450392767440, 16.17551501485623987692819329460, 17.14717750196177617596059546471, 17.753446811045424953337547711704, 18.60426061946069594186184634668, 20.24443110188818518002348367989, 20.99817640598974332687281958339, 22.22640301049393460076479793553, 23.03244395919811809041328453254, 23.684648437738583691078200965372, 24.5517608152089167995842799657, 25.96598156015072722602983338883, 26.35270836367493272530992883648

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