Properties

Degree 1
Conductor $ 5 \cdot 47 $
Sign $0.704 - 0.709i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(235\)    =    \(5 \cdot 47\)
\( \varepsilon \)  =  $0.704 - 0.709i$
motivic weight  =  \(0\)
character  :  $\chi_{235} (57, \cdot )$
Sato-Tate  :  $\mu(92)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 235,\ (0:\ ),\ 0.704 - 0.709i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.173287142 - 0.4884242340i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.173287142 - 0.4884242340i\)
\(L(\chi,1)\)  \(\approx\)  \(1.063870675 - 0.3574315021i\)
\(L(1,\chi)\)  \(\approx\)  \(1.063870675 - 0.3574315021i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

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