Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.303 + 0.952i$
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 + (−0.973 + 0.230i)3-s + 4-s + (0.998 − 0.0581i)5-s + (0.973 − 0.230i)6-s + (0.893 + 0.448i)7-s − 8-s + (0.893 − 0.448i)9-s + (−0.998 + 0.0581i)10-s + (0.998 + 0.0581i)11-s + (−0.973 + 0.230i)12-s + (0.0581 + 0.998i)13-s + (−0.893 − 0.448i)14-s + (−0.957 + 0.286i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯
L(s,χ)  = 1  − 2-s + (−0.973 + 0.230i)3-s + 4-s + (0.998 − 0.0581i)5-s + (0.973 − 0.230i)6-s + (0.893 + 0.448i)7-s − 8-s + (0.893 − 0.448i)9-s + (−0.998 + 0.0581i)10-s + (0.998 + 0.0581i)11-s + (−0.973 + 0.230i)12-s + (0.0581 + 0.998i)13-s + (−0.893 − 0.448i)14-s + (−0.957 + 0.286i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.303 + 0.952i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.303 + 0.952i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.303 + 0.952i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (466, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ -0.303 + 0.952i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5344831876 + 0.7312064463i$
$L(\frac12,\chi)$  $\approx$  $0.5344831876 + 0.7312064463i$
$L(\chi,1)$  $\approx$  0.6673251613 + 0.1801373116i
$L(1,\chi)$  $\approx$  0.6673251613 + 0.1801373116i

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

−17.94414996470616935190064722853, −17.61169599703594004992208911734, −17.24136536389820268910461194595, −16.44479961264037840213696805820, −15.95617296884936113544366335448, −14.81011451366079895178989162773, −14.36380268483517303776310444016, −13.37043330900355148062055776300, −12.63965690081562327783158116101, −11.84178099514597849621205448384, −11.22505326053111729296712903768, −10.66384647501577115330106149975, −10.00440079107112317466001749750, −9.47515458081429986254307331027, −8.42213422849322717178268449046, −7.86078612527586343061781380085, −6.95752461508496765845312865145, −6.37351895992586673808350058444, −5.80571005245823555789588689847, −4.977982514912952375867087969880, −4.08760101743567325792545536654, −2.81483980764741482441586197420, −1.7937776422092346477447977565, −1.41650121975425976400971362229, −0.42356492780751462727990894975, 1.08411787882085487595203117778, 1.87147895997123581321105887490, 2.15840164742678759732464214518, 3.78666916219147463990908445698, 4.51397907088077014493910976169, 5.55075899311679460995747420963, 6.053120656449235727413387903638, 6.70729230017985520023602440401, 7.32641414492008545990863423581, 8.61081834408168170560060299883, 8.96734333625112337417670863412, 9.64443898465470782393132940619, 10.4534340704632488572108307629, 10.99348556502860001335569642371, 11.65785393141687596515816237734, 12.163339817581845276965779465, 13.02322302629603782076341369483, 14.04199761118205403597481007314, 14.84821085159424991694087766224, 15.318418629611250975002909093605, 16.32656938796107954499843979649, 16.88457181319433699852392103373, 17.3855939727384118673107635004, 17.75885771208009803328316660979, 18.510494677535271160978002064907

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