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Properties

Label 1-235-235.232-r0-0-0
Degree $1$
Conductor $235$
Sign $-0.434 + 0.900i$
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  + (0.887 + 0.460i)2-s + (−0.942 + 0.334i)3-s + (0.576 + 0.816i)4-s + (−0.990 − 0.136i)6-s + (0.519 + 0.854i)7-s + (0.136 + 0.990i)8-s + (0.775 − 0.631i)9-s + (−0.962 − 0.269i)11-s + (−0.816 − 0.576i)12-s + (0.979 + 0.203i)13-s + (0.0682 + 0.997i)14-s + (−0.334 + 0.942i)16-s + (0.269 + 0.962i)17-s + (0.979 − 0.203i)18-s + (−0.917 − 0.398i)19-s + ⋯
L(s)  = 1  + (0.887 + 0.460i)2-s + (−0.942 + 0.334i)3-s + (0.576 + 0.816i)4-s + (−0.990 − 0.136i)6-s + (0.519 + 0.854i)7-s + (0.136 + 0.990i)8-s + (0.775 − 0.631i)9-s + (−0.962 − 0.269i)11-s + (−0.816 − 0.576i)12-s + (0.979 + 0.203i)13-s + (0.0682 + 0.997i)14-s + (−0.334 + 0.942i)16-s + (0.269 + 0.962i)17-s + (0.979 − 0.203i)18-s + (−0.917 − 0.398i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $-0.434 + 0.900i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (232, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ -0.434 + 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7841788835 + 1.248701853i\)
\(L(\frac12)\) \(\approx\) \(0.7841788835 + 1.248701853i\)
\(L(1)\) \(\approx\) \(1.095297645 + 0.7447859633i\)
\(L(1)\) \(\approx\) \(1.095297645 + 0.7447859633i\)

Euler product

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

−25.80833564658451915064455040311, −24.61728752334952111703248715060, −23.82000195280546225774783326719, −23.07064545828478370285308876034, −22.65844023894286192371344390978, −21.08461794997706966105562116928, −20.893780999506286857487986842502, −19.530923661522381821887158951378, −18.454571944728762278086040003998, −17.66808491002061547861792419592, −16.334873632743147025769592712214, −15.673981744463673902945145631976, −14.24214378902003294237810392981, −13.428635394459841997362922242951, −12.575827392806368421465525872248, −11.5757823848145427781671072752, −10.67876962852359816161703158879, −10.122469074110618349352340330156, −8.00480665864111796626620821256, −6.92829942661609209268797014822, −5.88304917809124281374037218584, −4.8851421717388449054909168737, −3.96631593429588593098227924043, −2.27549938010762887238575035529, −0.934276840907364431244752559553, 2.0038553476449998218678181355, 3.61136248053480589846342420192, 4.724547953573421361804509254913, 5.71541856482766320746872939218, 6.29232168536309498988775645625, 7.78483628088348693584945771012, 8.80957100801382485338893682762, 10.582384403457656014072603512886, 11.26648095274844824270354971044, 12.31326626230746738559250656506, 13.04979670192632707392254526757, 14.338968177228790805185369484958, 15.519340329519296095169534955623, 15.83243656643851993151501426517, 17.04693500432540836158649751526, 17.87572728125461171153116470005, 18.87132386616634223332143493152, 20.613765760790224797728200497458, 21.421140299521133217755240175022, 21.821989891263317607972588217259, 22.99964130620937639009686368394, 23.78472439905860389260529420733, 24.27055518273830027294491213498, 25.67436091999975999400365572614, 26.27245284528808720669185796608

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