Properties

Degree 1
Conductor 373
Sign $0.607 + 0.794i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.999 + 0.0337i)2-s + (−0.184 + 0.982i)3-s + (0.997 + 0.0675i)4-s + (0.839 − 0.543i)5-s + (−0.217 + 0.975i)6-s + (0.918 + 0.394i)7-s + (0.994 + 0.101i)8-s + (−0.931 − 0.363i)9-s + (0.857 − 0.514i)10-s + (−0.409 + 0.912i)11-s + (−0.250 + 0.968i)12-s + (−0.994 + 0.101i)13-s + (0.905 + 0.425i)14-s + (0.378 + 0.925i)15-s + (0.990 + 0.134i)16-s + (−0.440 − 0.897i)17-s + ⋯
L(s,χ)  = 1  + (0.999 + 0.0337i)2-s + (−0.184 + 0.982i)3-s + (0.997 + 0.0675i)4-s + (0.839 − 0.543i)5-s + (−0.217 + 0.975i)6-s + (0.918 + 0.394i)7-s + (0.994 + 0.101i)8-s + (−0.931 − 0.363i)9-s + (0.857 − 0.514i)10-s + (−0.409 + 0.912i)11-s + (−0.250 + 0.968i)12-s + (−0.994 + 0.101i)13-s + (0.905 + 0.425i)14-s + (0.378 + 0.925i)15-s + (0.990 + 0.134i)16-s + (−0.440 − 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.607 + 0.794i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.607 + 0.794i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(373\)
\( \varepsilon \)  =  $0.607 + 0.794i$
motivic weight  =  \(0\)
character  :  $\chi_{373} (4, \cdot )$
Sato-Tate  :  $\mu(186)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 373,\ (0:\ ),\ 0.607 + 0.794i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.429201259 + 1.199639077i$
$L(\frac12,\chi)$  $\approx$  $2.429201259 + 1.199639077i$
$L(\chi,1)$  $\approx$  1.974041911 + 0.6142759385i
$L(1,\chi)$  $\approx$  1.974041911 + 0.6142759385i

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

−24.30278546632241142216137729471, −23.791508060246912208666002225509, −22.78276651768470798339226022375, −21.8293057555668758063122688624, −21.34448645060894210060991465776, −20.11539800801335126744023000590, −19.35313210959461952983601348777, −18.26125422956689698672085145995, −17.376164782896997222290903614129, −16.72955920588962016016893474086, −15.15070536157642831827966066165, −14.440427477492431830115388069532, −13.55760390670877404230496920744, −13.16576246049095056887341837936, −11.903437087599171058266603964432, −11.08075895490308587929132315886, −10.41776542385061984011902752541, −8.6608852943191335189211779594, −7.36863022447107488063875705327, −6.89365046546492556633111922755, −5.58526244758372580654956840754, −5.14696119574621383944279011884, −3.38227495531122743980713955511, −2.33534402532102615010600352818, −1.4407931377692009700132223293, 1.85418439460395608870492705065, 2.77656392743423090842410952226, 4.346397402853740112776778414652, 5.066653368770712314162592595745, 5.48856058673997358892854029239, 6.905461912394375277762622958797, 8.1777741681416285995769220265, 9.490226902285081955382045172202, 10.189102860980995811757698525979, 11.37296681886425829827424184744, 12.09234545302121502752342732707, 13.11205320649649552613677716081, 14.245508439584791420833010717469, 14.83340327010456748801038410802, 15.64179619689254817040205439192, 16.7472826813077346702083563303, 17.27617907251092982831132859637, 18.43708434905178286964756696048, 20.264538433527428596881877134499, 20.56554605511110119525247367962, 21.23745210942127042776292185851, 22.18888225062473701943399557576, 22.63905725440619039952301716781, 23.888610297551365062086284147677, 24.70098044150076537911921669814

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