Properties

Degree 1
Conductor 569
Sign $-0.0468 + 0.998i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.0663 − 0.997i)2-s + (−0.666 + 0.745i)3-s + (−0.991 − 0.132i)4-s + (−0.787 + 0.616i)5-s + (0.699 + 0.714i)6-s + (−0.283 + 0.958i)7-s + (−0.197 + 0.980i)8-s + (−0.110 − 0.993i)9-s + (0.562 + 0.826i)10-s + (0.903 − 0.428i)11-s + (0.759 − 0.650i)12-s + (0.862 − 0.506i)13-s + (0.937 + 0.346i)14-s + (0.0663 − 0.997i)15-s + (0.964 + 0.262i)16-s + (0.996 − 0.0883i)17-s + ⋯
L(s,χ)  = 1  + (0.0663 − 0.997i)2-s + (−0.666 + 0.745i)3-s + (−0.991 − 0.132i)4-s + (−0.787 + 0.616i)5-s + (0.699 + 0.714i)6-s + (−0.283 + 0.958i)7-s + (−0.197 + 0.980i)8-s + (−0.110 − 0.993i)9-s + (0.562 + 0.826i)10-s + (0.903 − 0.428i)11-s + (0.759 − 0.650i)12-s + (0.862 − 0.506i)13-s + (0.937 + 0.346i)14-s + (0.0663 − 0.997i)15-s + (0.964 + 0.262i)16-s + (0.996 − 0.0883i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.0468 + 0.998i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.0468 + 0.998i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $-0.0468 + 0.998i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (101, \cdot )$
Sato-Tate  :  $\mu(71)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (0:\ ),\ -0.0468 + 0.998i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3939073397 + 0.4128239097i$
$L(\frac12,\chi)$  $\approx$  $0.3939073397 + 0.4128239097i$
$L(\chi,1)$  $\approx$  0.6599169907 + 0.02739264030i
$L(1,\chi)$  $\approx$  0.6599169907 + 0.02739264030i

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

−23.17751896904830981996474100885, −22.831275471982729321996337154, −21.66785821235301835151250702043, −20.42490200029196192195568744284, −19.4082786276860479686852794183, −18.85537014851216442287020095382, −17.77158535982287394304078423195, −16.86551719352913831505375783595, −16.59150903501045220591717261980, −15.74693153119249592625469519420, −14.47945826298542537382425785627, −13.790779302701672650389517840380, −12.74988425107340471897210572007, −12.2898799404621457313330376603, −11.14759416193591945546419727731, −10.05081292007907692250271212676, −8.75970796893705894279264841331, −8.053474644617837212386812794579, −7.0546976191850876554541612049, −6.57453712637881203881992988010, −5.47597266839290455096242352347, −4.320003018201348599621482026898, −3.79161158260528438221488262067, −1.47496690207316943394191489004, −0.37722489881642839321032861122, 1.30320347625267464527745636557, 3.07937592890318655272977511030, 3.51420003260574719542469408807, 4.54007794571538931023542859103, 5.714024707746978511458455328986, 6.41264640766513739026588712670, 8.17001949963390858992553282768, 8.929739406170885059389634821101, 9.95023835275997752143652262936, 10.68617934963513227488780511159, 11.63587431608953596378487597534, 11.91830472609297075018086971053, 12.93395284544318185714018141737, 14.320483529954775914175696755758, 14.97558560295795973147737591090, 15.85690454441102676476952768689, 16.75908833776676735781314328308, 17.91426415726500060940632590677, 18.530032118447518088113049908756, 19.36961073579627768564902764450, 20.10570154565463442414461119018, 21.33019693168552285908653694568, 21.68440628579553071813757547984, 22.539748519252688920921884620663, 23.14202110154223591489885299229

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