Properties

Label 1-235-235.88-r0-0-0
Degree $1$
Conductor $235$
Sign $0.405 - 0.914i$
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  + (−0.730 − 0.682i)2-s + (0.136 − 0.990i)3-s + (0.0682 + 0.997i)4-s + (−0.775 + 0.631i)6-s + (0.398 + 0.917i)7-s + (0.631 − 0.775i)8-s + (−0.962 − 0.269i)9-s + (−0.203 + 0.979i)11-s + (0.997 + 0.0682i)12-s + (0.519 − 0.854i)13-s + (0.334 − 0.942i)14-s + (−0.990 + 0.136i)16-s + (0.979 − 0.203i)17-s + (0.519 + 0.854i)18-s + (0.460 + 0.887i)19-s + ⋯
L(s)  = 1  + (−0.730 − 0.682i)2-s + (0.136 − 0.990i)3-s + (0.0682 + 0.997i)4-s + (−0.775 + 0.631i)6-s + (0.398 + 0.917i)7-s + (0.631 − 0.775i)8-s + (−0.962 − 0.269i)9-s + (−0.203 + 0.979i)11-s + (0.997 + 0.0682i)12-s + (0.519 − 0.854i)13-s + (0.334 − 0.942i)14-s + (−0.990 + 0.136i)16-s + (0.979 − 0.203i)17-s + (0.519 + 0.854i)18-s + (0.460 + 0.887i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $0.405 - 0.914i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ 0.405 - 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7713604394 - 0.5015599208i\)
\(L(\frac12)\) \(\approx\) \(0.7713604394 - 0.5015599208i\)
\(L(1)\) \(\approx\) \(0.7624953645 - 0.3644502351i\)
\(L(1)\) \(\approx\) \(0.7624953645 - 0.3644502351i\)

Euler product

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

−26.48469684415841742929116499273, −25.88835475220371960594434204523, −24.73118368976904999072877456478, −23.63695162222454858124726335171, −23.09592354025533708237122566368, −21.59852588439012975909995125631, −20.88378207262177523341041766328, −19.78869839264136842812517630013, −19.027353941466891312886975080268, −17.72780153694851437197116133882, −16.83932898019116025144736992569, −16.213793624444001552788636337914, −15.33983508230797299578421063000, −14.12973646660816571662765183352, −13.74766864680500557363753358585, −11.448891468457076406516650205448, −10.80486288093777973646147771109, −9.85792886035771360463735033203, −8.85247961344282308301422418105, −8.013246854985688278443874353941, −6.78347257971619808578304056338, −5.517071696980880674384275750, −4.53773283200648246577194255191, −3.17848795303046045265546378613, −1.142260393890745932399243442361, 1.15518959756390293858861857761, 2.28757291196282077490499661193, 3.26493264517083938534668095251, 5.14257896716159937842278593444, 6.52868686564472581803907765114, 7.82764568030175821997982377667, 8.30664935511727969793227563837, 9.53091540276849660560632781726, 10.64130285425270281717065917358, 12.00817944444662595960899393553, 12.28302963353338697802688600608, 13.373730713468382780148434194680, 14.63638625611248349809704260060, 15.80789654347264132634227577657, 17.181158322423534790518373776509, 17.94299663776499716260787450297, 18.60058280114472000508685233708, 19.33331512795590634382778021301, 20.590654889616926359356711252166, 20.94638066902664458987364526845, 22.51896010552299897385747906673, 23.13116582731836551684710941529, 24.734827333573661719497690339851, 25.16373220016120977053307239789, 25.95473730240126476080054175350

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