Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.893 - 0.448i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.766 − 0.642i)2-s + (0.5 + 0.866i)3-s + (0.173 − 0.984i)4-s + (0.866 + 0.5i)5-s + (0.939 + 0.342i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.984 − 0.173i)10-s + (−0.984 − 0.173i)11-s + (0.939 − 0.342i)12-s + (−0.5 − 0.866i)13-s + (0.173 + 0.984i)14-s + i·15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s,χ)  = 1  + (0.766 − 0.642i)2-s + (0.5 + 0.866i)3-s + (0.173 − 0.984i)4-s + (0.866 + 0.5i)5-s + (0.939 + 0.342i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.984 − 0.173i)10-s + (−0.984 − 0.173i)11-s + (0.939 − 0.342i)12-s + (−0.5 − 0.866i)13-s + (0.173 + 0.984i)14-s + i·15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.893 - 0.448i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.893 - 0.448i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.893 - 0.448i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (2312, \cdot )$
Sato-Tate  :  $\mu(36)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 4033,\ (0:\ ),\ -0.893 - 0.448i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.1271945285 - 0.5369584333i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.1271945285 - 0.5369584333i\)
\(L(\chi,1)\)  \(\approx\)  \(1.401753045 - 0.09638707829i\)
\(L(1,\chi)\)  \(\approx\)  \(1.401753045 - 0.09638707829i\)

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

−18.75890208435177282178480607169, −17.74680839275204005269802142994, −17.42276561731099658649187074998, −16.6195199954395116694113721255, −16.26362926779510979606636686388, −14.900334903982732430024621079033, −14.77657441761580060044876251435, −13.75752281522831658487833993133, −13.35685118741377243844266971838, −12.78339878208441988716120050307, −12.51162729659328584898793224590, −11.365236039627037676169199985475, −10.49456393437423050758998392299, −9.59375882385903706019101225169, −8.80278938801417772297105371012, −8.20046206654473504193015276163, −7.39541513062478948387340087793, −6.74613120292062974668322971744, −6.24282013857939023336229198608, −5.4782712262750733482514790489, −4.47115389921064233514116721523, −3.97490315534909171123605988482, −2.72442969257729487760620772249, −2.34224964471463136847576258030, −1.35449227839600983423116933293, 0.089449500214695164149665032174, 1.93364217507897728315373789523, 2.362770733289107033492935784493, 3.150826033261677262893901567751, 3.47267634280867301207861075631, 4.81045373245075512500595594400, 5.42312681168515172648216401611, 5.66490411676439913102551678313, 6.76618304618598377248266472412, 7.653902539177643992188578042360, 8.87346893329043433510508011437, 9.313786401086021139431956624006, 10.04197962801264478689350423947, 10.56869522114619446569508216630, 11.089907110194991293801442046259, 12.0589332614614401184506419714, 13.046047557765470176846731317, 13.21140584978686088442308329775, 14.12196374486639162905654290654, 14.85414304060339440760402008886, 15.16327538457085785712871192503, 15.90716341458567095466268542421, 16.547359366790735660491702609609, 17.71520677615243122849482894177, 18.316421435728610291066637243280

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