Properties

Degree 1
Conductor 179
Sign $-0.228 + 0.973i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.0176 + 0.999i)2-s + (−0.329 − 0.944i)3-s + (−0.999 + 0.0352i)4-s + (0.760 + 0.648i)5-s + (0.938 − 0.345i)6-s + (−0.123 + 0.992i)7-s + (−0.0529 − 0.998i)8-s + (−0.783 + 0.621i)9-s + (−0.635 + 0.772i)10-s + (−0.261 − 0.965i)11-s + (0.362 + 0.932i)12-s + (0.427 + 0.904i)13-s + (−0.994 − 0.105i)14-s + (0.362 − 0.932i)15-s + (0.997 − 0.0705i)16-s + (0.977 + 0.210i)17-s + ⋯
L(s,χ)  = 1  + (0.0176 + 0.999i)2-s + (−0.329 − 0.944i)3-s + (−0.999 + 0.0352i)4-s + (0.760 + 0.648i)5-s + (0.938 − 0.345i)6-s + (−0.123 + 0.992i)7-s + (−0.0529 − 0.998i)8-s + (−0.783 + 0.621i)9-s + (−0.635 + 0.772i)10-s + (−0.261 − 0.965i)11-s + (0.362 + 0.932i)12-s + (0.427 + 0.904i)13-s + (−0.994 − 0.105i)14-s + (0.362 − 0.932i)15-s + (0.997 − 0.0705i)16-s + (0.977 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.228 + 0.973i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.228 + 0.973i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(179\)
\( \varepsilon \)  =  $-0.228 + 0.973i$
motivic weight  =  \(0\)
character  :  $\chi_{179} (39, \cdot )$
Sato-Tate  :  $\mu(89)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 179,\ (0:\ ),\ -0.228 + 0.973i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5828412279 + 0.7354320192i$
$L(\frac12,\chi)$  $\approx$  $0.5828412279 + 0.7354320192i$
$L(\chi,1)$  $\approx$  0.8072452136 + 0.4549048881i
$L(1,\chi)$  $\approx$  0.8072452136 + 0.4549048881i

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

−27.48678674506550069924573610495, −26.22416189008521154726333760738, −25.58835277693635352938731620666, −23.78764791517877163946231066781, −22.945021654446802879055389985225, −22.1688323969248378705388390857, −20.97931953214707335183155717365, −20.57487549632455724395743575348, −19.82822382514346880193393449464, −18.13750298898179047298532568372, −17.340033072754107496580723628627, −16.62897911864772108995789433313, −15.18769787047193830350799334998, −13.986402082918535436757910580866, −13.04699863620531015462919948952, −12.075063367218681409925790642258, −10.67243069808359483720727981719, −10.12016148007209657165921184537, −9.33266992871091977890768599815, −8.02907102831600520711727088655, −5.981383285056572416639718826422, −4.83647803928109946341880859694, −4.06694576173869683776307283571, −2.608407198210166900938676307319, −0.838910378652298462055807280170, 1.71824585367994124387729962400, 3.321262324001824954615129035292, 5.49861840615204067585929481781, 5.96076565100022547368049736358, 6.89260273661397744641066946581, 8.1504880795611822109129285246, 9.06763695355581439096964750993, 10.47936523128614067846730415853, 11.896803515730624838779497100117, 12.96336577554338833295804336196, 13.98567163380901819060568114425, 14.5549417657723650709349205005, 16.05694630398337919328707461602, 16.86490988495541446786827759142, 18.04021558047519273205848581489, 18.5977421983608576410684007226, 19.20664519933880438574024282556, 21.50864752810328593926226778084, 21.863993903638663975101451547082, 23.19865439583489845062672796849, 23.77763460444103902309678881540, 24.9863644442212011395309318397, 25.41320756976656360342787509896, 26.28207448326906682034590621946, 27.56998848121041171742748112042

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