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Properties

Label 1-235-235.167-r0-0-0
Degree $1$
Conductor $235$
Sign $-0.164 - 0.986i$
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  + (−0.816 − 0.576i)2-s + (−0.631 + 0.775i)3-s + (0.334 + 0.942i)4-s + (0.962 − 0.269i)6-s + (−0.887 + 0.460i)7-s + (0.269 − 0.962i)8-s + (−0.203 − 0.979i)9-s + (−0.854 + 0.519i)11-s + (−0.942 − 0.334i)12-s + (−0.398 − 0.917i)13-s + (0.990 + 0.136i)14-s + (−0.775 + 0.631i)16-s + (−0.519 + 0.854i)17-s + (−0.398 + 0.917i)18-s + (0.682 − 0.730i)19-s + ⋯
L(s)  = 1  + (−0.816 − 0.576i)2-s + (−0.631 + 0.775i)3-s + (0.334 + 0.942i)4-s + (0.962 − 0.269i)6-s + (−0.887 + 0.460i)7-s + (0.269 − 0.962i)8-s + (−0.203 − 0.979i)9-s + (−0.854 + 0.519i)11-s + (−0.942 − 0.334i)12-s + (−0.398 − 0.917i)13-s + (0.990 + 0.136i)14-s + (−0.775 + 0.631i)16-s + (−0.519 + 0.854i)17-s + (−0.398 + 0.917i)18-s + (0.682 − 0.730i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $-0.164 - 0.986i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ -0.164 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1776323277 - 0.2097011058i\)
\(L(\frac12)\) \(\approx\) \(0.1776323277 - 0.2097011058i\)
\(L(1)\) \(\approx\) \(0.4416151359 - 0.04206948300i\)
\(L(1)\) \(\approx\) \(0.4416151359 - 0.04206948300i\)

Euler product

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

−26.519217139950892785636809265449, −25.59715944472542264848965782667, −24.664431307226033812798385906405, −23.88183381418240928824042824816, −23.13442359790292650146043742909, −22.257545406883773476979657545717, −20.704355843896642437749195760428, −19.55513825127327082799135222460, −18.859070214599342704102969118708, −18.177847203323740074776264908, −17.02334397242233319777603979947, −16.42801238856656945490604510754, −15.60424759457665388694370609598, −14.00579997833212891550754622902, −13.35904219005027140385665238594, −11.97583933720489134591255782836, −10.992563487460182480338211133203, −10.01868637282551825382674573653, −8.92222397612819860956161113633, −7.56993086853833621302965982371, −6.97899291811093077303257212358, −5.96750361938437311471497166408, −4.928883259601273165372541105794, −2.784145964764982636609368963941, −1.24798856336338151118000331261, 0.30003427999477557216660558180, 2.46800312249234947371300663201, 3.48875445915909388225192906844, 4.88968899832519153367591768650, 6.173759157913335155134377120252, 7.40779905671903477271994223146, 8.76880516359367785753685443046, 9.69998091401006067276555627133, 10.398366368321519225529339615086, 11.30553322722884712036295307201, 12.48755557866725052467482695629, 13.06123040683468137645900949888, 15.25529480602983980707365840937, 15.641389437703671036404614016481, 16.78351786379848774816875840750, 17.57525155370626331306036501934, 18.421751334752640873957785099438, 19.5376206118308542491616562248, 20.448594093182644517167711794151, 21.27708379857371504259608751311, 22.31698668595533014968422917924, 22.73309435953540158809755014858, 24.266089229476258369197571465524, 25.49673488890378970216919565669, 26.29098464605195050021238081134

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