Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.184 - 0.982i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + (−0.686 + 0.727i)3-s + 4-s + (0.835 + 0.549i)5-s + (0.686 − 0.727i)6-s + (−0.0581 + 0.998i)7-s − 8-s + (−0.0581 − 0.998i)9-s + (−0.835 − 0.549i)10-s + (−0.835 + 0.549i)11-s + (−0.686 + 0.727i)12-s + (0.835 + 0.549i)13-s + (0.0581 − 0.998i)14-s + (−0.973 + 0.230i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯
L(s,χ)  = 1  − 2-s + (−0.686 + 0.727i)3-s + 4-s + (0.835 + 0.549i)5-s + (0.686 − 0.727i)6-s + (−0.0581 + 0.998i)7-s − 8-s + (−0.0581 − 0.998i)9-s + (−0.835 − 0.549i)10-s + (−0.835 + 0.549i)11-s + (−0.686 + 0.727i)12-s + (0.835 + 0.549i)13-s + (0.0581 − 0.998i)14-s + (−0.973 + 0.230i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.184 - 0.982i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.184 - 0.982i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.184 - 0.982i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (770, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ -0.184 - 0.982i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1344188824 + 0.1619517792i$
$L(\frac12,\chi)$  $\approx$  $-0.1344188824 + 0.1619517792i$
$L(\chi,1)$  $\approx$  0.4696321188 + 0.2937801196i
$L(1,\chi)$  $\approx$  0.4696321188 + 0.2937801196i

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

−17.958437264013364741802663899113, −17.36635415319515180872496002507, −16.769476350873550227046322855541, −16.27400780143996122793445331628, −15.72408064701999685849277621176, −14.457784150812839788245325952643, −13.71114243746872001949159505608, −12.97100711855018493433851386396, −12.698911480445630341087649296778, −11.57385230733434272956169799300, −10.77252822410839712673891965672, −10.60941018784110149144739712355, −9.775607055864704766866490446847, −8.759822062829928973475232635604, −8.30831101469119348086805881610, −7.4347576882338980669450708566, −6.810476968561322171301523933691, −6.20071206971469520174725167709, −5.34985998273908786937403905673, −4.81942530795611923384621245157, −3.24264475823479058995670555367, −2.60083618741914341376267069085, −1.39531681512107035557345629176, −1.07686705715344230855993627277, −0.095722391764933535354515807755, 1.528397452294939474010794608410, 2.07853387988580230952626650136, 3.03679426000657120224807730466, 3.809709640270172850187874806130, 5.12711372254077020260719894779, 5.712373099659211368768785205, 6.31929230676631810569542706137, 6.84013291365617424751432155970, 8.048729106190222315648762123161, 8.65240095626740885672608438909, 9.54931212444337547543825172850, 9.86767243428270282281376234225, 10.61776650254598145415599049397, 11.13518903744539675640060204548, 11.851382311315918473445052841117, 12.61317993205723958516200881524, 13.359360614569584777229412333, 14.670403135251813411114898011062, 15.160642239401268537626241838006, 15.47332832016304361901861258, 16.568584324014137626426589571567, 16.86283247009426178623224215342, 17.66610688398253403219076256533, 18.24454731550389360724051689552, 18.72261990441716512093402737978

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