Properties

Degree 1
Conductor 83
Sign $0.874 - 0.484i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.114 + 0.993i)2-s + (−0.409 − 0.912i)3-s + (−0.973 + 0.227i)4-s + (−0.0383 + 0.999i)5-s + (0.859 − 0.511i)6-s + (0.606 − 0.795i)7-s + (−0.338 − 0.941i)8-s + (−0.665 + 0.746i)9-s + (−0.997 + 0.0765i)10-s + (−0.927 − 0.373i)11-s + (0.606 + 0.795i)12-s + (0.543 − 0.839i)13-s + (0.859 + 0.511i)14-s + (0.927 − 0.373i)15-s + (0.896 − 0.443i)16-s + (0.988 − 0.152i)17-s + ⋯
L(s,χ)  = 1  + (0.114 + 0.993i)2-s + (−0.409 − 0.912i)3-s + (−0.973 + 0.227i)4-s + (−0.0383 + 0.999i)5-s + (0.859 − 0.511i)6-s + (0.606 − 0.795i)7-s + (−0.338 − 0.941i)8-s + (−0.665 + 0.746i)9-s + (−0.997 + 0.0765i)10-s + (−0.927 − 0.373i)11-s + (0.606 + 0.795i)12-s + (0.543 − 0.839i)13-s + (0.859 + 0.511i)14-s + (0.927 − 0.373i)15-s + (0.896 − 0.443i)16-s + (0.988 − 0.152i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.874 - 0.484i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.874 - 0.484i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $0.874 - 0.484i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (60, \cdot )$
Sato-Tate  :  $\mu(82)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (1:\ ),\ 0.874 - 0.484i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.143407372 - 0.2954417697i$
$L(\frac12,\chi)$  $\approx$  $1.143407372 - 0.2954417697i$
$L(\chi,1)$  $\approx$  0.9172976869 + 0.09053228764i
$L(1,\chi)$  $\approx$  0.9172976869 + 0.09053228764i

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

−31.09667896238918969532082239501, −29.20221980523824060497363496025, −28.51450968451017839664971197316, −27.83087773717826568108786429883, −26.96446939568869809748153592072, −25.548633854854599389864828623504, −23.858552184023908450749947915496, −23.09709129325240313505096853251, −21.61342326265761190492081890633, −21.07123911833471377949663080797, −20.35750126823795825529800344364, −18.77750853886087943916937795610, −17.693389987167019487040201607296, −16.46438820309351988441144014803, −15.27070189211430372199434619196, −13.92624167095836695532437094888, −12.38125249965532629552386088611, −11.710104330464977547300895403528, −10.41318896771949558766047825092, −9.27073699663180789049489439812, −8.36140159301025145235985113139, −5.43375057825816354364679930672, −4.91303288047175718775254055099, −3.40336377756832791894374582116, −1.453266897161340600119121312597, 0.6381380672325382518483991070, 3.11714348417879759412822186927, 5.11499865599070482217836557875, 6.28490541656471630094150897957, 7.496885782449581618920625387249, 8.05099277119169819527115777639, 10.25044855590536998789758343082, 11.40505819208090909544755680665, 13.11716141409922164549466535838, 13.8466020172542963449578927414, 14.9608504461509313467750561346, 16.32123266761129952216640917073, 17.57959044647672300930171721690, 18.21112387103943708056256148516, 19.16511781072245973807025370407, 20.98570327454558286750665598359, 22.56702246933555259727132629881, 23.156588128328325023050864301868, 24.02223740080996897964118495088, 25.10222229383905908370061626812, 26.17780251551850082475636964779, 27.029561363790030462601050177972, 28.32576242605456570109162747945, 29.89334692878530255380143256805, 30.44074534659209440455021062391

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