Properties

Degree 1
Conductor 569
Sign $-0.188 - 0.982i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.980 + 0.197i)2-s + (0.165 + 0.986i)3-s + (0.921 + 0.387i)4-s + (0.408 + 0.912i)5-s + (−0.0331 + 0.999i)6-s + (−0.650 − 0.759i)7-s + (0.826 + 0.562i)8-s + (−0.945 + 0.325i)9-s + (0.219 + 0.975i)10-s + (−0.515 − 0.856i)11-s + (−0.230 + 0.973i)12-s + (−0.999 + 0.0221i)13-s + (−0.487 − 0.873i)14-s + (−0.833 + 0.553i)15-s + (0.699 + 0.714i)16-s + (−0.262 − 0.964i)17-s + ⋯
L(s,χ)  = 1  + (0.980 + 0.197i)2-s + (0.165 + 0.986i)3-s + (0.921 + 0.387i)4-s + (0.408 + 0.912i)5-s + (−0.0331 + 0.999i)6-s + (−0.650 − 0.759i)7-s + (0.826 + 0.562i)8-s + (−0.945 + 0.325i)9-s + (0.219 + 0.975i)10-s + (−0.515 − 0.856i)11-s + (−0.230 + 0.973i)12-s + (−0.999 + 0.0221i)13-s + (−0.487 − 0.873i)14-s + (−0.833 + 0.553i)15-s + (0.699 + 0.714i)16-s + (−0.262 − 0.964i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.188 - 0.982i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.188 - 0.982i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $-0.188 - 0.982i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (47, \cdot )$
Sato-Tate  :  $\mu(568)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (1:\ ),\ -0.188 - 0.982i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.05081372280 - 0.06146972187i$
$L(\frac12,\chi)$  $\approx$  $0.05081372280 - 0.06146972187i$
$L(\chi,1)$  $\approx$  1.290459700 + 0.6370346172i
$L(1,\chi)$  $\approx$  1.290459700 + 0.6370346172i

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

−23.43640744907792967790298696078, −22.51231245734988602896888528994, −21.62965859774140542674530229824, −20.86656018215698369158505654485, −19.89183333653967248473612543786, −19.43014077919351322549227507451, −18.463293848557612592459035049136, −17.25643868316007625819747238538, −16.679441131214440695715565369441, −15.24164613141754956775588375876, −14.93264225879521546009590773534, −13.55559934795512427391277617728, −12.9987688910989033164338220880, −12.4048097981027766283951451676, −11.91004093208085611453448550448, −10.45292822665674212261311193319, −9.46141542020275279320483273968, −8.41108624803304590147426003361, −7.313607348580764045692119102271, −6.39756852974805344522012978035, −5.55727673250490871657669471385, −4.77666519447649744675461023824, −3.36436654166384698654596434435, −2.18404425344410117575596344248, −1.71281384178675816691607962413, 0.0111286355269346989844813886, 2.461525216004640783517240407788, 3.00131131463518531573505394737, 3.95993848504278464679591913166, 4.942649025696713178485764313090, 5.88386419847173100648311216440, 6.8432908549220132537586214491, 7.650228292887346452143072310789, 9.091407495885767189361907644991, 10.13337504115629696012728018370, 10.83921625195044607343880245283, 11.42882982022604607984032675037, 12.954841972306460360379208861079, 13.57671106969777098431875024428, 14.51626613862110933490962727186, 14.94794626347975340186471112990, 16.02810009876446998347747013168, 16.63932329407007810122367515048, 17.41352564958640165776996535352, 18.89113303379514622246212167685, 19.73066431180393791592947844657, 20.59241885532104382715677425256, 21.390326600897657538710923118586, 22.07352753681938056451554094954, 22.6450322482119003985796648634

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