Properties

Degree 1
Conductor 283
Sign $-0.809 - 0.587i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.645 + 0.763i)2-s + (0.296 + 0.955i)3-s + (−0.166 + 0.986i)4-s + (0.824 + 0.565i)5-s + (−0.538 + 0.842i)6-s + (0.359 + 0.933i)7-s + (−0.860 + 0.509i)8-s + (−0.824 + 0.565i)9-s + (0.100 + 0.994i)10-s + (−0.166 − 0.986i)11-s + (−0.991 + 0.133i)12-s + (−0.538 + 0.842i)13-s + (−0.480 + 0.876i)14-s + (−0.296 + 0.955i)15-s + (−0.944 − 0.328i)16-s + (−0.480 − 0.876i)17-s + ⋯
L(s,χ)  = 1  + (0.645 + 0.763i)2-s + (0.296 + 0.955i)3-s + (−0.166 + 0.986i)4-s + (0.824 + 0.565i)5-s + (−0.538 + 0.842i)6-s + (0.359 + 0.933i)7-s + (−0.860 + 0.509i)8-s + (−0.824 + 0.565i)9-s + (0.100 + 0.994i)10-s + (−0.166 − 0.986i)11-s + (−0.991 + 0.133i)12-s + (−0.538 + 0.842i)13-s + (−0.480 + 0.876i)14-s + (−0.296 + 0.955i)15-s + (−0.944 − 0.328i)16-s + (−0.480 − 0.876i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.809 - 0.587i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.809 - 0.587i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(283\)
\( \varepsilon \)  =  $-0.809 - 0.587i$
motivic weight  =  \(0\)
character  :  $\chi_{283} (67, \cdot )$
Sato-Tate  :  $\mu(94)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 283,\ (1:\ ),\ -0.809 - 0.587i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(-0.9047545352 + 2.786532069i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(-0.9047545352 + 2.786532069i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8083027965 + 1.512465187i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8083027965 + 1.512465187i\)

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

−24.64755274158589648718819839496, −23.868508838942082983310069992067, −23.06216983284833808564658313214, −22.15585715922935162292650694025, −20.87659945293387872041319322695, −20.2560065597163794269548432776, −19.7962087833840682819447518127, −18.395510505556773141463980152577, −17.72443992108614996065554976749, −16.84454396366404708874698645526, −15.04019195689736657399636520618, −14.31664155799733168389236522952, −13.40615111676075151072408415116, −12.75809909603034084100805374889, −12.0801603124770149994525092848, −10.606949477131472313213939802913, −9.91222196526306130748751720105, −8.637124052604018686775844567379, −7.4170786955938749768600237680, −6.25741006941771386240479235147, −5.20352142304235635480021564801, −4.089038231612152244963055921744, −2.57384123203848041467932410988, −1.67338424883703932327961158294, −0.66084592776325349515700002053, 2.47309018502932998497523010470, 3.13064599243095724266772807188, 4.67685761217214243184558016395, 5.41258913831476314276164348769, 6.35271293213334202481582780547, 7.65145533747126943797819356770, 9.01130651716431214203557952532, 9.37594373471313302246070990149, 11.04626245082102893475157529333, 11.73266250919673665277581429040, 13.356141592885655397731925943944, 14.06999391113896050121099842135, 14.74654027502415342320829442562, 15.70569234889222618187986364797, 16.38649495334082448531748630190, 17.524524692080100965621544303273, 18.29949581016524792574383968205, 19.63587549778120655397092665919, 21.082260969311709382446076187401, 21.67479360775410275780275565253, 21.92407355279830802419574034005, 23.03412879740858622137903370975, 24.326650206115653817789184909106, 25.01245996558428740952837204242, 25.814603143193395910627725463571

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