Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.0675 + 0.997i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.939 − 0.342i)2-s + (0.973 + 0.230i)3-s + (0.766 + 0.642i)4-s + (0.893 − 0.448i)5-s + (−0.835 − 0.549i)6-s + (−0.835 − 0.549i)7-s + (−0.5 − 0.866i)8-s + (0.893 + 0.448i)9-s + (−0.993 + 0.116i)10-s + (−0.993 − 0.116i)11-s + (0.597 + 0.802i)12-s + (−0.0581 + 0.998i)13-s + (0.597 + 0.802i)14-s + (0.973 − 0.230i)15-s + (0.173 + 0.984i)16-s + (−0.939 − 0.342i)17-s + ⋯
L(s,χ)  = 1  + (−0.939 − 0.342i)2-s + (0.973 + 0.230i)3-s + (0.766 + 0.642i)4-s + (0.893 − 0.448i)5-s + (−0.835 − 0.549i)6-s + (−0.835 − 0.549i)7-s + (−0.5 − 0.866i)8-s + (0.893 + 0.448i)9-s + (−0.993 + 0.116i)10-s + (−0.993 − 0.116i)11-s + (0.597 + 0.802i)12-s + (−0.0581 + 0.998i)13-s + (0.597 + 0.802i)14-s + (0.973 − 0.230i)15-s + (0.173 + 0.984i)16-s + (−0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0675 + 0.997i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0675 + 0.997i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.0675 + 0.997i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (2010, \cdot )$
Sato-Tate  :  $\mu(27)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.0675 + 0.997i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6438043378 + 0.6016623117i$
$L(\frac12,\chi)$  $\approx$  $0.6438043378 + 0.6016623117i$
$L(\chi,1)$  $\approx$  0.8658510091 + 0.03067643183i
$L(1,\chi)$  $\approx$  0.8658510091 + 0.03067643183i

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

−18.24205067498872588094019832444, −17.967237336857393524362856701, −17.024624978422131029989898707127, −16.247492947702142987838573741269, −15.48386392218659620578847528626, −14.99748368159458456703515490563, −14.52460454225195231123334939168, −13.45961501054815431562591753295, −12.94234368239988781063562735654, −12.40313538888614648382253778211, −11.019901789596695143438106580802, −10.41077525740406914335445474983, −9.90827123697646357151325690518, −9.23521826968024605079150250411, −8.64154178701364287511366952877, −7.88115080699887687237973923972, −7.28443707398928228436091953555, −6.32533774851626928256091402857, −6.01442590823069848314758409037, −5.074030426094625638526486070234, −3.7007631068321939330442897727, −2.68871125408616239387374057746, −2.42228845489502583009180156755, −1.66692597188040252815288093111, −0.275985858140029864222712200173, 1.08179510528628425491357536870, 2.04499921253571053276306177087, 2.503035429981222443320192107945, 3.34346856361985990832321868634, 4.17646028028608740819334843168, 5.06458180329221823510455540630, 6.20235660829846072930866211456, 7.092970662591695568323913880866, 7.36709433000758538471001907058, 8.647033101816374538366495522156, 8.92013743284548966805864362705, 9.54769683618734482694011376170, 10.16867474698115976446377743706, 10.72660231510292475963730253158, 11.58308910682734844186772746281, 12.74887971220574590304479764404, 13.23427796111993778406393491575, 13.5449418051082515468647079271, 14.53457061262212629889358623275, 15.60468764399083932225452476538, 15.90049566785680080152356794011, 16.68899339782710012523455572349, 17.22881398810010892439139099720, 18.1119635565232076235465152028, 18.653143420099740956667182647746

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