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Properties

Label 1-235-235.38-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: $\chi_{235} (38, \cdot )$
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

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

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