Properties

Degree 1
Conductor $ 5 \cdot 19 $
Sign $0.934 - 0.356i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.939 + 0.342i)2-s + (−0.173 − 0.984i)3-s + (0.766 + 0.642i)4-s + (0.173 − 0.984i)6-s + (0.5 − 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (−0.173 + 0.984i)13-s + (0.766 − 0.642i)14-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s − 18-s + (−0.939 − 0.342i)21-s + (−0.173 − 0.984i)22-s + ⋯
L(s,χ)  = 1  + (0.939 + 0.342i)2-s + (−0.173 − 0.984i)3-s + (0.766 + 0.642i)4-s + (0.173 − 0.984i)6-s + (0.5 − 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (−0.173 + 0.984i)13-s + (0.766 − 0.642i)14-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s − 18-s + (−0.939 − 0.342i)21-s + (−0.173 − 0.984i)22-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.934 - 0.356i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.934 - 0.356i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(95\)    =    \(5 \cdot 19\)
\( \varepsilon \)  =  $0.934 - 0.356i$
motivic weight  =  \(0\)
character  :  $\chi_{95} (74, \cdot )$
Sato-Tate  :  $\mu(18)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 95,\ (0:\ ),\ 0.934 - 0.356i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.571692994 - 0.2899569907i$
$L(\frac12,\chi)$  $\approx$  $1.571692994 - 0.2899569907i$
$L(\chi,1)$  $\approx$  1.578159458 - 0.1664993896i
$L(1,\chi)$  $\approx$  1.578159458 - 0.1664993896i

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

−30.42335578433688237400055247932, −29.24936978158029641501085661698, −27.984666376661910173003331734941, −27.72142283242071668724777519826, −25.89989367323361328665123223969, −25.01366945189344378631744064477, −23.68009492002541015124627109655, −22.63744711921003035556577046608, −21.90724193126677750095452628855, −20.81764171262074136620427107979, −20.26840475087088801799861468145, −18.65755477301279328288338449285, −17.26393434299206033266490828115, −15.71295794635145761170139075103, −15.20978044189647898871208352133, −14.17790823292114283677645583338, −12.58695125626857444353258068218, −11.68532749793297873508842668555, −10.497172697587405053648741958646, −9.52017565619962028900128537477, −7.714465745071805576471901776856, −5.698043794023226827710570017105, −5.11564430202304506665285428883, −3.67639323412865879741081794284, −2.30929494499540452280877220539, 1.76632440623587475050293720066, 3.46136663112393954914399035571, 5.05282497212988865566950424601, 6.30175843895685207422135691777, 7.39785994433572317732510582948, 8.32858434780013019276724817197, 10.72958036012125169169724160348, 11.71619309511336784301089594127, 12.82961738240976584427396797173, 13.9054061034512150883355944988, 14.4736576849830076244645743652, 16.34021950856856501891259533399, 16.99737708059424991065365713823, 18.36651169793960011929371083784, 19.618010660093505292210271669927, 20.765224934983019215067838981795, 21.84284817638075760753100048494, 23.175044943641470856167296800851, 23.85767516458756251124195992460, 24.45993883390187835403735074497, 25.77432886185773253347660027443, 26.66076109625168621466219535293, 28.44699283325503850813464615500, 29.57925368913104635764749910676, 30.098273250519332063081516386406

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