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Properties

Label 1-4235-4235.108-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.0474 + 0.998i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (−0.999 − 0.00951i)2-s + (0.207 + 0.978i)3-s + (0.999 + 0.0190i)4-s + (−0.198 − 0.980i)6-s + (−0.999 − 0.0285i)8-s + (−0.913 + 0.406i)9-s + (0.189 + 0.981i)12-s + (0.996 − 0.0855i)13-s + (0.999 + 0.0380i)16-s + (−0.836 + 0.548i)17-s + (0.917 − 0.398i)18-s + (0.964 + 0.263i)19-s + (0.618 − 0.786i)23-s + (−0.179 − 0.983i)24-s + (−0.997 + 0.0760i)26-s + (−0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.999 − 0.00951i)2-s + (0.207 + 0.978i)3-s + (0.999 + 0.0190i)4-s + (−0.198 − 0.980i)6-s + (−0.999 − 0.0285i)8-s + (−0.913 + 0.406i)9-s + (0.189 + 0.981i)12-s + (0.996 − 0.0855i)13-s + (0.999 + 0.0380i)16-s + (−0.836 + 0.548i)17-s + (0.917 − 0.398i)18-s + (0.964 + 0.263i)19-s + (0.618 − 0.786i)23-s + (−0.179 − 0.983i)24-s + (−0.997 + 0.0760i)26-s + (−0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0474 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0474 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.0474 + 0.998i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.0474 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7665188856 + 0.8037764401i\)
\(L(\frac12)\) \(\approx\) \(0.7665188856 + 0.8037764401i\)
\(L(1)\) \(\approx\) \(0.7167719647 + 0.2825674087i\)
\(L(1)\) \(\approx\) \(0.7167719647 + 0.2825674087i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.999 - 0.00951i)T \)
3 \( 1 + (0.207 + 0.978i)T \)
13 \( 1 + (0.996 - 0.0855i)T \)
17 \( 1 + (-0.836 + 0.548i)T \)
19 \( 1 + (0.964 + 0.263i)T \)
23 \( 1 + (0.618 - 0.786i)T \)
29 \( 1 + (-0.774 - 0.633i)T \)
31 \( 1 + (0.683 + 0.730i)T \)
37 \( 1 + (-0.603 - 0.797i)T \)
41 \( 1 + (0.736 + 0.676i)T \)
43 \( 1 + (-0.281 - 0.959i)T \)
47 \( 1 + (-0.917 - 0.398i)T \)
53 \( 1 + (0.0380 + 0.999i)T \)
59 \( 1 + (0.953 + 0.299i)T \)
61 \( 1 + (-0.861 + 0.508i)T \)
67 \( 1 + (0.0950 + 0.995i)T \)
71 \( 1 + (-0.362 + 0.931i)T \)
73 \( 1 + (0.768 - 0.640i)T \)
79 \( 1 + (0.991 - 0.132i)T \)
83 \( 1 + (0.717 - 0.696i)T \)
89 \( 1 + (-0.888 + 0.458i)T \)
97 \( 1 + (-0.336 + 0.941i)T \)
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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

−18.23352975610247411496705628015, −17.731804672178496850063804630487, −17.068389698216911870158499295336, −16.275878215693942763101781154941, −15.607181327917304526909568212693, −14.97129996977658058380637767125, −14.02113728373578421092742416786, −13.4065394615322673263589517235, −12.76275678342122315684110784368, −11.81471698784499823283293688979, −11.32508755850561066502489062210, −10.85564375684522706438599775866, −9.581060755012512343873894178313, −9.250180467981908093109507526796, −8.41052764658981428215554976013, −7.86679713448461644649480232155, −7.091077878963420994573384935037, −6.59625634288213543278285105868, −5.853307107917925551057413107680, −4.99906908878022849202752519568, −3.52796551574254161488897529337, −2.997977068610270748881725317026, −2.03496712321053829950656294369, −1.37329691977261167018991176132, −0.54941254869729964254691272643, 0.81793488138494126156725726907, 1.87502815220405475507893577992, 2.74352208887250642940908306816, 3.49090015127296341721935894563, 4.215221370720000627315315599819, 5.276774109639213761129539306480, 5.98108986790405733287160239608, 6.727831195005701442521431421046, 7.65856000707935576723303332287, 8.4446553314902678188054464913, 8.86797734639684085609642508511, 9.53273683966194086823656520295, 10.33100588106908641773140490788, 10.79842885429380979816971010897, 11.400572002663911344622513366531, 12.13335012987483653595890364850, 13.154736616890613593990292780789, 13.92245069240756768362628652478, 14.8284705419302524515112953597, 15.30197630956608647269751089036, 16.05695301682407114008735235834, 16.368954815052912228076305318106, 17.23085513706079985520259728719, 17.78097702640356230763957673083, 18.51826087939411240521162092119

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