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Properties

Label 1-235-235.87-r0-0-0
Degree $1$
Conductor $235$
Sign $0.681 + 0.731i$
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  + (0.136 − 0.990i)2-s + (−0.519 − 0.854i)3-s + (−0.962 − 0.269i)4-s + (−0.917 + 0.398i)6-s + (−0.997 − 0.0682i)7-s + (−0.398 + 0.917i)8-s + (−0.460 + 0.887i)9-s + (−0.682 + 0.730i)11-s + (0.269 + 0.962i)12-s + (0.816 − 0.576i)13-s + (−0.203 + 0.979i)14-s + (0.854 + 0.519i)16-s + (−0.730 + 0.682i)17-s + (0.816 + 0.576i)18-s + (−0.334 + 0.942i)19-s + ⋯
L(s)  = 1  + (0.136 − 0.990i)2-s + (−0.519 − 0.854i)3-s + (−0.962 − 0.269i)4-s + (−0.917 + 0.398i)6-s + (−0.997 − 0.0682i)7-s + (−0.398 + 0.917i)8-s + (−0.460 + 0.887i)9-s + (−0.682 + 0.730i)11-s + (0.269 + 0.962i)12-s + (0.816 − 0.576i)13-s + (−0.203 + 0.979i)14-s + (0.854 + 0.519i)16-s + (−0.730 + 0.682i)17-s + (0.816 + 0.576i)18-s + (−0.334 + 0.942i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $0.681 + 0.731i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ 0.681 + 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1622496290 + 0.07057773535i\)
\(L(\frac12)\) \(\approx\) \(0.1622496290 + 0.07057773535i\)
\(L(1)\) \(\approx\) \(0.4669615181 - 0.3220583282i\)
\(L(1)\) \(\approx\) \(0.4669615181 - 0.3220583282i\)

Euler product

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

−26.20215809455512180240212942808, −25.4695112727733197787251835527, −24.09507915990078298693038362647, −23.39722567683886708429527354392, −22.53330560519202370135455386022, −21.784799557924305040962056243097, −20.964279634167883537590372030526, −19.474595988472396864995538970045, −18.37963794455079449462522938282, −17.46554860908868409577113944308, −16.34638312664139996692094344175, −15.92546845444452578313183007515, −15.19081947591009822801452676715, −13.76646122970849146550902581237, −13.10332699090027305358501903747, −11.67339993168094215749284278581, −10.56126075406683799664376991857, −9.32638051083816249326579295947, −8.79130360613133960892003231213, −7.15337006211797235806857281387, −6.16650934849562911673614425797, −5.363800921904881775535999975844, −4.138570414254131960137107126508, −3.161326955181723078561938830701, −0.12927375473696335193755518999, 1.581411389696768326886422826, 2.737413190589344237080119145862, 4.06338999636000716547779266632, 5.49674615398216672346300073846, 6.40200979469507230077238177868, 7.81230998517578786283061628016, 8.94407342486861427675190571898, 10.36241847537555248980343222987, 10.85304760432715227814280017844, 12.30845425773288912610963460059, 12.807201668719172530158332717435, 13.46414740545229022046172553453, 14.79022419182331374730585435173, 16.175335747380177590958431532108, 17.3183896934131765515115292429, 18.28511834970002054789117899344, 18.835461237255077100858432344, 19.89656772665682972312190288525, 20.59780754925516116145327789059, 21.943682528229790704535168517507, 22.803758787303700526715442121074, 23.226665960451353458031489300, 24.29678029333507492880773745459, 25.590909389993243589481511727797, 26.32473389904156520801903727709

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