↑

Properties

Label 1-235-235.159-r0-0-0
Degree $1$
Conductor $235$
Sign $0.698 - 0.715i$
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  + (0.334 + 0.942i)2-s + (−0.203 − 0.979i)3-s + (−0.775 + 0.631i)4-s + (0.854 − 0.519i)6-s + (0.576 − 0.816i)7-s + (−0.854 − 0.519i)8-s + (−0.917 + 0.398i)9-s + (0.460 − 0.887i)11-s + (0.775 + 0.631i)12-s + (−0.682 + 0.730i)13-s + (0.962 + 0.269i)14-s + (0.203 − 0.979i)16-s + (−0.460 − 0.887i)17-s + (−0.682 − 0.730i)18-s + (−0.0682 − 0.997i)19-s + ⋯
L(s)  = 1  + (0.334 + 0.942i)2-s + (−0.203 − 0.979i)3-s + (−0.775 + 0.631i)4-s + (0.854 − 0.519i)6-s + (0.576 − 0.816i)7-s + (−0.854 − 0.519i)8-s + (−0.917 + 0.398i)9-s + (0.460 − 0.887i)11-s + (0.775 + 0.631i)12-s + (−0.682 + 0.730i)13-s + (0.962 + 0.269i)14-s + (0.203 − 0.979i)16-s + (−0.460 − 0.887i)17-s + (−0.682 − 0.730i)18-s + (−0.0682 − 0.997i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $0.698 - 0.715i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ 0.698 - 0.715i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9879403839 - 0.4161579412i\)
\(L(\frac12)\) \(\approx\) \(0.9879403839 - 0.4161579412i\)
\(L(1)\) \(\approx\) \(1.029729039 - 0.04067425550i\)
\(L(1)\) \(\approx\) \(1.029729039 - 0.04067425550i\)

Euler product

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

−26.85625253199731034761326715836, −25.50486497554957504202601041307, −24.46530465123970647253102163330, −23.12198572558768028187505501304, −22.543302676112712450281440360073, −21.56869114982798329900358759175, −21.06915144950766131923960923907, −20.02276199171649171179449747426, −19.25930323956586725906318628446, −17.74165922765113338394501982279, −17.40297191506349799537896918032, −15.64502617108678950952118969495, −14.91783283225674521567907584953, −14.23743697021110545635642906445, −12.608832916058916028607472164853, −11.98196010031893249070117041068, −10.946426722850456740697959038631, −10.04262627483296681625521394508, −9.22111772981657716145908243221, −8.14255316642084949580601743909, −6.06193919400228499600676484547, −5.10551420349864906341562697576, −4.26008373758021020960417607737, −3.04304711723689465261192372701, −1.76691273575236163323754055016, 0.73963903938859858907546642567, 2.65610805641849289184279973037, 4.28305799100584210073146697212, 5.287016739970034064567871680262, 6.67129412383716529631553579455, 7.11174169924512236958396626386, 8.26234774883608464755612232937, 9.169965895545066318636569199603, 10.99495507427715422113760793227, 11.90915092535815627764136837960, 13.0363404525427021135795696948, 14.02183770287970247226928773646, 14.32036101494979985897048334701, 15.88914678723709547483650115365, 16.95961461089358091225919519205, 17.402172588080028830120126169208, 18.48933346953840781216292765992, 19.38741124373214840780949286022, 20.60654025852527679544806298178, 21.906311444130344745193517976527, 22.66355802341539026222733533268, 23.74853808968382081090339057597, 24.25534950119555734274683774696, 24.83953513991578957334135196167, 26.107127983341993717118396466461

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