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Properties

Label 1-235-235.57-r0-0-0
Degree $1$
Conductor $235$
Sign $0.704 - 0.709i$
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  + (−0.398 − 0.917i)2-s + (0.997 + 0.0682i)3-s + (−0.682 + 0.730i)4-s + (−0.334 − 0.942i)6-s + (−0.979 + 0.203i)7-s + (0.942 + 0.334i)8-s + (0.990 + 0.136i)9-s + (0.775 − 0.631i)11-s + (−0.730 + 0.682i)12-s + (−0.269 + 0.962i)13-s + (0.576 + 0.816i)14-s + (−0.0682 − 0.997i)16-s + (0.631 − 0.775i)17-s + (−0.269 − 0.962i)18-s + (0.854 + 0.519i)19-s + ⋯
L(s)  = 1  + (−0.398 − 0.917i)2-s + (0.997 + 0.0682i)3-s + (−0.682 + 0.730i)4-s + (−0.334 − 0.942i)6-s + (−0.979 + 0.203i)7-s + (0.942 + 0.334i)8-s + (0.990 + 0.136i)9-s + (0.775 − 0.631i)11-s + (−0.730 + 0.682i)12-s + (−0.269 + 0.962i)13-s + (0.576 + 0.816i)14-s + (−0.0682 − 0.997i)16-s + (0.631 − 0.775i)17-s + (−0.269 − 0.962i)18-s + (0.854 + 0.519i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $0.704 - 0.709i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ 0.704 - 0.709i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.173287142 - 0.4884242340i\)
\(L(\frac12)\) \(\approx\) \(1.173287142 - 0.4884242340i\)
\(L(1)\) \(\approx\) \(1.063870675 - 0.3574315021i\)
\(L(1)\) \(\approx\) \(1.063870675 - 0.3574315021i\)

Euler product

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

−26.0051960788569734477871626015, −25.62229302073824414243758729640, −24.82798448488331224253683711539, −23.88898226578764937227787048986, −22.78458495686868002046281851023, −22.023614284561343189599580373867, −20.473421912352287094254720809933, −19.57951837616032625115687766432, −19.11215417876979773980907549984, −17.83484314846547808200740608205, −16.99349144008712906667022377238, −15.74462300191651084142388423209, −15.208532031573829389794377563877, −14.17395836271672059669767127207, −13.31927842178524370206130803144, −12.371348160172751945407599254263, −10.32732620095302934735994774677, −9.65615129898572119669267302385, −8.81084946974019979054934271520, −7.607842532095383704158204700757, −6.96130456867070393050278838092, −5.698932647616514304759560118386, −4.21821751138354318715437943410, −3.083405239686924017312523308883, −1.27274525620816310115198519632, 1.30421145222130133313780291885, 2.77578837640033422167541277551, 3.4461148337942817464604645566, 4.64850127884422559603507999631, 6.588956241358126863481727776226, 7.77130527096502803504484892439, 9.017846555508372771276045715543, 9.42338869934467870067140670568, 10.4568815242873759647079481136, 11.8711702595686662277670091682, 12.5980314714781239012510490748, 13.86092394805232221736595332714, 14.28646149575995852238537822409, 16.03886278354590944021812525167, 16.60525587396027343147173761866, 18.14281564420258517683278092685, 19.11110136687827094829768706511, 19.42481035965771184750944734952, 20.48411416257626756505228587680, 21.346746745154596477745299985897, 22.13958684963726166074510945431, 23.1064080220598178760809168070, 24.70914833667499424807376651610, 25.328717825191760438225683904257, 26.58703090531976981178698410341

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