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Properties

Label 1-235-235.198-r0-0-0
Degree $1$
Conductor $235$
Sign $0.319 + 0.947i$
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  + (0.398 + 0.917i)2-s + (−0.997 − 0.0682i)3-s + (−0.682 + 0.730i)4-s + (−0.334 − 0.942i)6-s + (0.979 − 0.203i)7-s + (−0.942 − 0.334i)8-s + (0.990 + 0.136i)9-s + (0.775 − 0.631i)11-s + (0.730 − 0.682i)12-s + (0.269 − 0.962i)13-s + (0.576 + 0.816i)14-s + (−0.0682 − 0.997i)16-s + (−0.631 + 0.775i)17-s + (0.269 + 0.962i)18-s + (0.854 + 0.519i)19-s + ⋯
L(s)  = 1  + (0.398 + 0.917i)2-s + (−0.997 − 0.0682i)3-s + (−0.682 + 0.730i)4-s + (−0.334 − 0.942i)6-s + (0.979 − 0.203i)7-s + (−0.942 − 0.334i)8-s + (0.990 + 0.136i)9-s + (0.775 − 0.631i)11-s + (0.730 − 0.682i)12-s + (0.269 − 0.962i)13-s + (0.576 + 0.816i)14-s + (−0.0682 − 0.997i)16-s + (−0.631 + 0.775i)17-s + (0.269 + 0.962i)18-s + (0.854 + 0.519i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $0.319 + 0.947i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (198, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ 0.319 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9244627308 + 0.6638339194i\)
\(L(\frac12)\) \(\approx\) \(0.9244627308 + 0.6638339194i\)
\(L(1)\) \(\approx\) \(0.9275313276 + 0.4585375379i\)
\(L(1)\) \(\approx\) \(0.9275313276 + 0.4585375379i\)

Euler product

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

−26.535340051215615864657967850238, −24.654049758408398133952035853748, −24.10352708702488644069415375471, −23.07308365469204411789720383210, −22.30996360614807206589767026277, −21.55439241311952165453164830368, −20.68652249887097910460671776747, −19.73052248373347199525922407274, −18.33662877864074925080277032938, −18.00443046697531845287370464257, −16.863975198604360839053853363389, −15.59790283607127500698239872681, −14.4996880978634592799764149652, −13.59081945848119710296091381077, −12.26446849300234563958956847725, −11.646456820169542353017396263587, −11.01204025436440471242034579616, −9.79722542223352805775814652244, −8.8722500308417328322287261280, −7.10329363801382326544465449093, −5.95122984819972960476849092869, −4.7115035394450710679657495425, −4.24228478882210546166770419615, −2.29553597286511308818616230945, −1.12025040503829497839994287769, 1.215213213454391757561091708499, 3.56323242529377559018594780337, 4.67625129089038547903851986354, 5.633909099701381294907804357585, 6.458745204301403047705324771297, 7.65968903997942071060286559196, 8.51767436359385735499628992605, 10.04000839451299465506358054423, 11.28062026371220578673415598820, 12.069043893920867687920846446346, 13.22495497778047399430815907003, 14.1292978484100501098156778573, 15.243109140129901799626498055529, 16.09450920633612127985358954136, 17.123097433771398350086417394263, 17.69122642299433533575246761312, 18.46951804069794531878403011347, 20.01117266325384015949689004788, 21.44819240034025305478121687536, 21.90725168574794852705947614420, 22.992305836474799615103198564287, 23.65798118300495728391467107694, 24.550902436895385441589182715843, 25.10609438658679160764213463353, 26.64947993308232518686344795998

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