Properties

Degree 1
Conductor 17
Sign $0.825 + 0.563i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.825 + 0.563i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.825 + 0.563i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(17\)
\( \varepsilon \)  =  $0.825 + 0.563i$
motivic weight  =  \(0\)
character  :  $\chi_{17} (6, \cdot )$
Sato-Tate  :  $\mu(16)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 17,\ (1:\ ),\ 0.825 + 0.563i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.840176969 + 0.5683876217i$
$L(\frac12,\chi)$  $\approx$  $1.840176969 + 0.5683876217i$
$L(\chi,1)$  $\approx$  1.601018356 + 0.3927743947i
$L(1,\chi)$  $\approx$  1.601018356 + 0.3927743947i

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

−41.356594899673041909870744826531, −39.18406921738597757547237818760, −38.71030887733946944097871582558, −37.33290540260283670869258826027, −36.19564635020667518488833083851, −33.781296017591045360792929367389, −32.6779217949817317264432997110, −31.21204385185497133225527849782, −30.49334852836046047606501340170, −28.96846965737148135531620437605, −27.03524747256711623408936707860, −25.99112907078403410387067861606, −23.86813184871099079942706211527, −22.54222792750195125378914065321, −21.08612712929105260798887509290, −19.82757322441046247619632825665, −18.71331050288806291606370812484, −15.74841843590598773856371535688, −14.42418557286325225650823050816, −13.26577436173584387516768142941, −11.02131628444003477640997179942, −9.75854802093811489627795191217, −7.162390560226882352945119354540, −4.27218557517400748548055345135, −2.76709448198816416655662751888, 3.16711075027993145263853882, 5.432051278064953406097622403, 7.71972133544652081438822773681, 8.89291284661477505556827442935, 12.36356835450350969236681067984, 13.25926508920288807793805024749, 15.08948043180965154777665971151, 16.1370625505243793736745381069, 18.29658711683568404583581042989, 20.16805365293293228901629694862, 21.4457288751630009563315276924, 23.37256169213771993885544259767, 24.67536749472359774974969489193, 25.4534845382737813313063178993, 27.09526202085607845668823357721, 29.19819432418985816304961338771, 31.03356287429880353240626572945, 31.71635953614421929616383063506, 32.72539385669846024139506052155, 34.780228724629630593423218653551, 35.64013803892604917260216774223, 37.13295793077064265061505583342, 38.91874155992885368810518458846, 40.27339812826001888244580245722, 41.65323987285134220259858973271

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