Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.993 + 0.112i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.173 + 0.984i)2-s + (0.973 + 0.230i)3-s + (−0.939 − 0.342i)4-s + (0.835 + 0.549i)5-s + (−0.396 + 0.918i)6-s + (−0.0581 + 0.998i)7-s + (0.5 − 0.866i)8-s + (0.893 + 0.448i)9-s + (−0.686 + 0.727i)10-s + (−0.686 − 0.727i)11-s + (−0.835 − 0.549i)12-s + (0.0581 − 0.998i)13-s + (−0.973 − 0.230i)14-s + (0.686 + 0.727i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯
L(s,χ)  = 1  + (−0.173 + 0.984i)2-s + (0.973 + 0.230i)3-s + (−0.939 − 0.342i)4-s + (0.835 + 0.549i)5-s + (−0.396 + 0.918i)6-s + (−0.0581 + 0.998i)7-s + (0.5 − 0.866i)8-s + (0.893 + 0.448i)9-s + (−0.686 + 0.727i)10-s + (−0.686 − 0.727i)11-s + (−0.835 − 0.549i)12-s + (0.0581 − 0.998i)13-s + (−0.973 − 0.230i)14-s + (0.686 + 0.727i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.993 + 0.112i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.993 + 0.112i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.993 + 0.112i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (1824, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ -0.993 + 0.112i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1161420114 + 2.049698214i$
$L(\frac12,\chi)$  $\approx$  $0.1161420114 + 2.049698214i$
$L(\chi,1)$  $\approx$  0.9450015311 + 0.9332438575i
$L(1,\chi)$  $\approx$  0.9450015311 + 0.9332438575i

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

−18.130212453501847394793670315553, −17.585829844519632496062964380984, −17.1537742999232673326785232596, −16.14506198581169358286489505076, −15.376971472841032281447060231143, −14.20659169280336468502742893484, −13.96394073316888210654893633609, −13.303936969143453403512191571, −12.78716886502558980350178241010, −12.212552558075875361315601276136, −11.09766951834479717255651179626, −10.37746599635795815494914527221, −9.81399771262042828409361153034, −9.284145750135702546310723397998, −8.50702559311617400402861308458, −7.98369035805961391532056711804, −6.99521953918169323688311171037, −6.339962262791927953058113767935, −4.90249573918101439001130248255, −4.40778401151337660185286894893, −3.8616311198010465934699903242, −2.64175528726232607809099687470, −2.16724297230155291434976081937, −1.52518922631434824840866641323, −0.51873374849558356748668260232, 1.22756947507189651951379973838, 2.328351161796911435791350586431, 2.90606138897853674942703780129, 3.72822722516237412735983644886, 4.819926817780123541192835347889, 5.63252671803976121271894146250, 5.96833513939735776155329922174, 6.97018232938757482951180212368, 7.73090614192854360767252203036, 8.41713968971122516485776608216, 8.88364836045469295084208727111, 9.71297931255531698906359331796, 10.19364216381047768219522029892, 10.879201295525372888237108058212, 12.17215299992229657038586649160, 13.13825571236803644314228729800, 13.4612055885425301227819806971, 14.18231017350555622595518980515, 14.7606818219226214447428729840, 15.488734300586406552468516189592, 15.790068900813387699495337870852, 16.588240406321029800706589653293, 17.57487003848567559990132530290, 18.29356018191318245261444153721, 18.47986449375469278603815822119

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