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Properties

Label 1-235-235.169-r0-0-0
Degree $1$
Conductor $235$
Sign $0.161 + 0.986i$
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  + (0.775 + 0.631i)2-s + (0.917 + 0.398i)3-s + (0.203 + 0.979i)4-s + (0.460 + 0.887i)6-s + (0.334 − 0.942i)7-s + (−0.460 + 0.887i)8-s + (0.682 + 0.730i)9-s + (−0.576 + 0.816i)11-s + (−0.203 + 0.979i)12-s + (0.0682 − 0.997i)13-s + (0.854 − 0.519i)14-s + (−0.917 + 0.398i)16-s + (0.576 + 0.816i)17-s + (0.0682 + 0.997i)18-s + (−0.990 − 0.136i)19-s + ⋯
L(s)  = 1  + (0.775 + 0.631i)2-s + (0.917 + 0.398i)3-s + (0.203 + 0.979i)4-s + (0.460 + 0.887i)6-s + (0.334 − 0.942i)7-s + (−0.460 + 0.887i)8-s + (0.682 + 0.730i)9-s + (−0.576 + 0.816i)11-s + (−0.203 + 0.979i)12-s + (0.0682 − 0.997i)13-s + (0.854 − 0.519i)14-s + (−0.917 + 0.398i)16-s + (0.576 + 0.816i)17-s + (0.0682 + 0.997i)18-s + (−0.990 − 0.136i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $0.161 + 0.986i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ 0.161 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.826233875 + 1.551677792i\)
\(L(\frac12)\) \(\approx\) \(1.826233875 + 1.551677792i\)
\(L(1)\) \(\approx\) \(1.743141920 + 0.9532577280i\)
\(L(1)\) \(\approx\) \(1.743141920 + 0.9532577280i\)

Euler product

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

−25.809496492175114359258169596666, −25.00735834513960077363156602139, −24.06264537552172189325989328701, −23.53201042684137722790379183961, −22.09403885904825124144265300364, −21.157800952551540202287960435116, −20.860890549272589022889418618772, −19.443189828718184185906567896, −18.871964633824743431565656470134, −18.1981957047416202386097734067, −16.30029447971402251079655285511, −15.26389581139894606621618229601, −14.470349970179502413705177598700, −13.671142452677438347853215511630, −12.734724563188814672796600848259, −11.84290918251338286797699821475, −10.8273038729427352183704704650, −9.394952942836502293350552653498, −8.69766041928059606017119400955, −7.270429775841197893665898319483, −6.02802740261886750249211228480, −4.89332416960293155382603823175, −3.509117132034014603428656557474, −2.579081835358422343156218912083, −1.53702923475425619064466026077, 2.13216819839441272937888765433, 3.4222981201400193008714428674, 4.32727986743288928040573879342, 5.299469431292891046684168333723, 6.899539642478361357325263747039, 7.802806373500830517263696756445, 8.51108240279818057296334775030, 10.07954744925058398828229391232, 10.91145339446220356709290211138, 12.7306242604288890875910223350, 13.131529838651213743599636783682, 14.392225280162935285233830412064, 14.92227151867795802642206186592, 15.79916732961049863694424120413, 16.91422623641797121144973962739, 17.73156601773237369570587896095, 19.2308407877977259595773935698, 20.44610129978991681838898974618, 20.79005757567327088000133188716, 21.81077600717475122497470762308, 22.98924094176219896702758678354, 23.61225099877084666823221756373, 24.762590588095231973109695865817, 25.52588358283657455712341947605, 26.24445112197281353897444497101

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