Properties

Degree 1
Conductor 17
Sign $-0.974 - 0.226i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.707 + 0.707i)2-s + (−0.382 + 0.923i)3-s i·4-s + (−0.923 − 0.382i)5-s + (−0.382 − 0.923i)6-s + (−0.923 + 0.382i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.923 − 0.382i)10-s + (0.382 + 0.923i)11-s + (0.923 + 0.382i)12-s + i·13-s + (0.382 − 0.923i)14-s + (0.707 − 0.707i)15-s − 16-s + ⋯
L(s,χ)  = 1  + (−0.707 + 0.707i)2-s + (−0.382 + 0.923i)3-s i·4-s + (−0.923 − 0.382i)5-s + (−0.382 − 0.923i)6-s + (−0.923 + 0.382i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.923 − 0.382i)10-s + (0.382 + 0.923i)11-s + (0.923 + 0.382i)12-s + i·13-s + (0.382 − 0.923i)14-s + (0.707 − 0.707i)15-s − 16-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.974 - 0.226i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.974 - 0.226i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(17\)
\( \varepsilon \)  =  $-0.974 - 0.226i$
motivic weight  =  \(0\)
character  :  $\chi_{17} (5, \cdot )$
Sato-Tate  :  $\mu(16)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 17,\ (1:\ ),\ -0.974 - 0.226i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.03785959933 + 0.3304265109i$
$L(\frac12,\chi)$  $\approx$  $-0.03785959933 + 0.3304265109i$
$L(\chi,1)$  $\approx$  0.3364536872 + 0.3041433867i
$L(1,\chi)$  $\approx$  0.3364536872 + 0.3041433867i

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

−40.08933506006515232808569501134, −39.211164042317428677088923901181, −37.79132475493121548469594362424, −36.250160631826820748051406908228, −35.203900470717652026601521520725, −34.49681110980341751107496182124, −31.88421685697285491575892731058, −30.30074573595674422592446311993, −29.59897888664145406627834667602, −28.13248118072899166640715101029, −26.77719057098203746301071286072, −25.27940750172787845585789905595, −23.38375312098535499146615790182, −22.14392892474512541309571125350, −19.73391427861055133875059216657, −19.19445133339640635093713708536, −17.64120167489276756286468707560, −16.12148499173592123748792240253, −13.333629792910303461114000633001, −11.95784778691279866741936781380, −10.63861745698672798356651503622, −8.29867536775569339857252816986, −6.8194345706205735510214385089, −3.232520385351869734905576312524, −0.39131461302555943968992017527, 4.45150760621423341536424083440, 6.521377799321187945309143225667, 8.72077279114639747851188082385, 10.05473234794703825384217167606, 11.95487896113766360394246049370, 14.84726188590983910559904346551, 15.98742108078147625047328159174, 16.93645045266218797071450118487, 18.947383980442873732651471005600, 20.32897469362333379148347235501, 22.51119148271240252445779939656, 23.63587325283735955260546353974, 25.4886707480401936511815226321, 26.7494894689177514473207280004, 27.946773560061612964642362317315, 28.78234775346773357309072737393, 31.54570671160629637629440255225, 32.60751041348255207926129951700, 33.914463546227765267034293948626, 35.142633592076596907751364156085, 36.166647137494799468204170725424, 38.22430974545368939352183426367, 38.76340003441337217120280084163, 40.6851019417213447809555920791, 42.35923171085841664792465386996

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