Properties

Degree 1
Conductor 17
Sign $0.615 + 0.788i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.615 + 0.788i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.615 + 0.788i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(17\)
\( \varepsilon \)  =  $0.615 + 0.788i$
motivic weight  =  \(0\)
character  :  $\chi_{17} (13, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 17,\ (0:\ ),\ 0.615 + 0.788i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3883506224 + 0.1894872842i$
$L(\frac12,\chi)$  $\approx$  $0.3883506224 + 0.1894872842i$
$L(\chi,1)$  $\approx$  0.5927817455 + 0.1936151527i
$L(1,\chi)$  $\approx$  0.5927817455 + 0.1936151527i

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

−41.39825263368656834810158053634, −40.123878538062289210737176714428, −38.3670491737333797985499135149, −36.9750433058488300935484291589, −35.83300407184436397685818801208, −35.14296567168541579991627230428, −33.60034550680774720031512265464, −31.5696493152975137686651010417, −30.10578496434668135472201722168, −28.522996391203429046323481231197, −28.00152292947194639614103841511, −25.62481024876169433791299063204, −24.91955176224272177116750484121, −23.53276396731361401358774471776, −20.964356893460047508874269655244, −19.6158544746063732711902498863, −18.30609178276429653661407253897, −17.122130095672791600965426780671, −15.40203147631056025262030921185, −12.86274574324480776117856226684, −11.65632735313783772341642685163, −9.19390267662336140187995090804, −7.97395225057117766202200498016, −6.02106312276084192766295517421, −1.89456288883640063451868939480, 3.38764301980058083148824334443, 6.466503328866556453198478281136, 8.51273109083455322179148702773, 10.41283347696993241919453381820, 11.03897117769565301368588396339, 14.27914098300065745959487204470, 15.854658931004722037734299863, 17.09285647216109024420336241365, 18.76358864398668875311125644434, 20.28073895145710467217235457053, 21.6437758792531046000649108338, 23.41778677783755363216066933962, 25.69146946873353330579060148931, 26.59620981521950272346296366303, 27.49201032500673839848802587111, 29.14964593988215408206112800369, 30.445439668547355789741455455925, 32.676185030466947999125108619010, 33.76074984228369908790641957324, 34.81856497845666396290521055952, 36.601442066436625569348311303850, 37.78303357163219485745268114579, 38.64551375754757981318534880869, 39.95820595043727429615498631813, 42.537824722990093368896081175906

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