Properties

Degree 1
Conductor 569
Sign $0.997 + 0.0741i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.984 + 0.176i)2-s + (0.930 − 0.367i)3-s + (0.937 − 0.346i)4-s + (−0.197 + 0.980i)5-s + (−0.850 + 0.525i)6-s + (−0.240 − 0.970i)7-s + (−0.862 + 0.506i)8-s + (0.730 − 0.683i)9-s + (0.0221 − 0.999i)10-s + (−0.132 + 0.991i)11-s + (0.745 − 0.666i)12-s + (−0.154 − 0.988i)13-s + (0.408 + 0.912i)14-s + (0.176 + 0.984i)15-s + (0.759 − 0.650i)16-s + (0.283 + 0.958i)17-s + ⋯
L(s,χ)  = 1  + (−0.984 + 0.176i)2-s + (0.930 − 0.367i)3-s + (0.937 − 0.346i)4-s + (−0.197 + 0.980i)5-s + (−0.850 + 0.525i)6-s + (−0.240 − 0.970i)7-s + (−0.862 + 0.506i)8-s + (0.730 − 0.683i)9-s + (0.0221 − 0.999i)10-s + (−0.132 + 0.991i)11-s + (0.745 − 0.666i)12-s + (−0.154 − 0.988i)13-s + (0.408 + 0.912i)14-s + (0.176 + 0.984i)15-s + (0.759 − 0.650i)16-s + (0.283 + 0.958i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.997 + 0.0741i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.997 + 0.0741i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $0.997 + 0.0741i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (50, \cdot )$
Sato-Tate  :  $\mu(284)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (0:\ ),\ 0.997 + 0.0741i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.200266053 + 0.04455386126i$
$L(\frac12,\chi)$  $\approx$  $1.200266053 + 0.04455386126i$
$L(\chi,1)$  $\approx$  0.9526082652 + 0.02343103976i
$L(1,\chi)$  $\approx$  0.9526082652 + 0.02343103976i

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

−23.56826852571722709749712299223, −21.82535447723337537146676799919, −21.3310580017163776420639806178, −20.785516962127407722682182540689, −19.55656528716515546792171335798, −19.35852480992508091943623131981, −18.509258569423714921406386306695, −17.36992340222935137882605861179, −16.2583309478587425721031946892, −15.97047422373853514869280645489, −15.14870302348081565318603470546, −13.889356197383029863141406373191, −12.99332816843982610091224416052, −11.93439470897625588166069435484, −11.27040605554802166111137344289, −9.91951928413176458603497130825, −9.096485342446383385774034192557, −8.810987565213488274229660494379, −7.94180864993265839340958528415, −6.86339031785786219922178249370, −5.54237898043778042174286726134, −4.376821401644774801879108483240, −3.06466812978636751365536483616, −2.35174332884996813969927122775, −1.011597656380517317239235037, 1.0364427416738197013294908301, 2.31132780759719144039329877030, 3.12013358930390463396212121193, 4.23314759470692802215679253320, 6.18898184168141693049350670152, 6.85628374833984574848686878101, 7.818239216488146873345149530697, 8.07644769411532622982753061298, 9.57827395576198087784893859384, 10.24246923222597769306205531824, 10.761191833766799704720967338212, 12.230957143914374782424446318592, 13.00612836497316702120730361177, 14.371728981272256995777762989556, 14.803939058507941483917051948174, 15.54855940462126769687711180857, 16.67542075476996487943802067494, 17.73154736873579368428532861431, 18.17517874648207646358096882194, 19.39892962492702690545379638306, 19.50433655498294373728870644120, 20.54995158680527354438654114020, 21.15754900378000335544949533699, 22.75299892646423749384276203602, 23.30981140582124400227911411102

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