Properties

Label 1-235-235.132-r0-0-0
Degree $1$
Conductor $235$
Sign $0.955 - 0.294i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $0.955 - 0.294i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ 0.955 - 0.294i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.332241558 - 0.3506197354i\)
\(L(\frac12)\) \(\approx\) \(2.332241558 - 0.3506197354i\)
\(L(1)\) \(\approx\) \(1.940526057 - 0.2459987176i\)
\(L(1)\) \(\approx\) \(1.940526057 - 0.2459987176i\)

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 \( 1 + (-0.631 - 0.775i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.15065450991619299306677222718, −25.24996708848846953189550162413, −24.26602807703153805992338631043, −23.746644097794256643187484170198, −23.00457015721734441877830112036, −21.63092873548729960384037533788, −20.65023542759828507076517547863, −20.2337010280457239427573945246, −18.62614878631159306943820039780, −17.88891086477163008021031926291, −16.87926473664774178987713207389, −15.63233384870234856517730723682, −14.79207571663987571064695263392, −13.67705090098086152710032445188, −13.50019630443059174038907253939, −12.02125598580661908775096782103, −11.4016223245895737478580867540, −9.58422644399682126420760061488, −8.204732962270613749920537946000, −7.57219759298874464022242948755, −6.705699023045762944400446959541, −5.33219504571885484072562859873, −4.23194475581575763380978248668, −2.92968768513135564170769057071, −1.73177425769461957617728093687, 1.7403141400389957970619810253, 2.955508590513771716250080090780, 3.86855122423940968211925326332, 5.17385973482945478582286050381, 5.755606843732466373562078445396, 7.809851784412982867775590878041, 8.652186617453222118390040284940, 10.15677164044176978459692214484, 10.64868640297736056001689110711, 11.79180125388371304485448746907, 12.949538887985385731265035988549, 13.95858637165671304015052991203, 14.74697962838871667687895857754, 15.5078225894013779828030922121, 16.40243388056580436760190497070, 18.10247811239974127734452503097, 18.94516744658700455899115696466, 20.13988636039659309868252639931, 20.78648498719189227025682548737, 21.45574197740898367844664672778, 22.256062141791595431160623893580, 23.32277113091487258839350333229, 24.37413670772225601900741455004, 25.118091874452750464779824521018, 26.26981603515303268657693472796

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