Properties

Degree 1
Conductor 17
Sign $-0.139 - 0.990i$
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

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.139 - 0.990i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.139 - 0.990i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(17\)
\( \varepsilon \)  =  $-0.139 - 0.990i$
motivic weight  =  \(0\)
character  :  $\chi_{17} (14, \cdot )$
Sato-Tate  :  $\mu(16)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 17,\ (1:\ ),\ -0.139 - 0.990i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9311494398 - 1.071252131i$
$L(\frac12,\chi)$  $\approx$  $0.9311494398 - 1.071252131i$
$L(\chi,1)$  $\approx$  1.026557040 - 0.7114857552i
$L(1,\chi)$  $\approx$  1.026557040 - 0.7114857552i

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

−41.48946198471389814419910491101, −40.31591073775791236539514048537, −39.335895821080204099454304634676, −37.86254223053293509339853159624, −35.67345877705159363372997709813, −34.38300182012545449675241340593, −33.415988976217645313609009981041, −32.59142479550074655351488845417, −30.40688913720403841833202134516, −29.64646318212120248292668307050, −27.402890689538725399263918583460, −26.263220833794432448847595709004, −24.55088429721233190497002767122, −22.9160878133856775638251956415, −22.35274006712663527817276174439, −20.73601382461062255678490327801, −17.89162486314073795673833439090, −16.96255332088990513809409072366, −15.258727862890474415073123448188, −13.88726765877519019925207643961, −11.900561085775338517174447415529, −10.26525357790367285198891490709, −7.29713881290272697352366687991, −5.87613513547569449746020404318, −3.96733015716415479332555752191, 1.554269601485073702014516769817, 4.7768875978404875933929745300, 6.12454143257586315359542021887, 9.28558905646388458326732102136, 11.45953084658036273651653779807, 12.34958587404643874845085677239, 13.91997128725079869171238053916, 16.10500204401020727088348424817, 17.87907737409314949954753945566, 19.43553024609536487202111720969, 21.29930234154393680288336591490, 22.19566986157050046937089496603, 23.98511641809200456207271133468, 24.67870197679031250096714000512, 27.78030082620755135134436060800, 28.520825290047799633209856230773, 29.69276975087506697806333371884, 31.138367874699192222369633610588, 32.569625581196314344038725156453, 33.82607093372941793115291640616, 35.48248964918470882152905094983, 36.96290295731723323815929585104, 38.44645407061706632432161588107, 39.87164407935438441616508614389, 40.68959831521997260475436352804

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