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Properties

Label 1-235-235.67-r0-0-0
Degree $1$
Conductor $235$
Sign $-0.959 - 0.282i$
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  + (−0.136 − 0.990i)2-s + (0.519 − 0.854i)3-s + (−0.962 + 0.269i)4-s + (−0.917 − 0.398i)6-s + (0.997 − 0.0682i)7-s + (0.398 + 0.917i)8-s + (−0.460 − 0.887i)9-s + (−0.682 − 0.730i)11-s + (−0.269 + 0.962i)12-s + (−0.816 − 0.576i)13-s + (−0.203 − 0.979i)14-s + (0.854 − 0.519i)16-s + (0.730 + 0.682i)17-s + (−0.816 + 0.576i)18-s + (−0.334 − 0.942i)19-s + ⋯
L(s)  = 1  + (−0.136 − 0.990i)2-s + (0.519 − 0.854i)3-s + (−0.962 + 0.269i)4-s + (−0.917 − 0.398i)6-s + (0.997 − 0.0682i)7-s + (0.398 + 0.917i)8-s + (−0.460 − 0.887i)9-s + (−0.682 − 0.730i)11-s + (−0.269 + 0.962i)12-s + (−0.816 − 0.576i)13-s + (−0.203 − 0.979i)14-s + (0.854 − 0.519i)16-s + (0.730 + 0.682i)17-s + (−0.816 + 0.576i)18-s + (−0.334 − 0.942i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $-0.959 - 0.282i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ -0.959 - 0.282i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1629127033 - 1.129322847i\)
\(L(\frac12)\) \(\approx\) \(0.1629127033 - 1.129322847i\)
\(L(1)\) \(\approx\) \(0.6680480708 - 0.8179073563i\)
\(L(1)\) \(\approx\) \(0.6680480708 - 0.8179073563i\)

Euler product

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

−26.66368851804494387902031603393, −25.704506856896630638242586500735, −25.08869314773685326409907524544, −24.00897238847465046745083234719, −23.157521457195633156648306196, −22.067178250049220534392647178723, −21.205028895200156690668866437083, −20.32722384894592298900697516578, −19.06102504490428038476949175037, −18.11597606228628229871647603236, −17.05367183350930209537875368451, −16.29704123056445166336358348191, −15.20616431501545846964993798050, −14.600699797540606769215154425350, −13.89947879413686420520830083461, −12.51207316129941051004918328016, −11.01843347507255801537306370030, −9.863206756292895852427212782589, −9.178629497380494066773826602918, −7.89620840921793951444159209431, −7.40766775169092017295291184964, −5.508546830518741723551046739647, −4.87969469039340679065392940307, −3.79800122652061421858499697006, −2.02634380690736297809175637556, 0.84374968021127352740938208038, 2.19000608816976722428267635412, 3.05655316942580069096039361994, 4.53450166208213718621684276530, 5.79074346656738052963007843655, 7.62345168268868935389822873988, 8.16888061330754973165312254816, 9.21194994802174836589560809898, 10.56022803815975924622061880662, 11.40093932032142263640891935804, 12.52586206127794722447603341199, 13.17278692757051313064754125449, 14.27392114591432503985477421159, 14.94576530044798010861097911808, 16.87988285566924035181399581437, 17.7578576582666289550882772717, 18.5056406497904382995863261425, 19.35351619818915901633639513086, 20.19710253738460129559880656392, 21.04987725070211844318745040970, 21.83696670447782465169008196240, 23.19578501174224160791118276108, 23.94985693682358717436640168670, 24.805827250910655244756565385448, 26.121657775175902343083125425914

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