Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.765 - 0.643i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.765 - 0.643i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.765 - 0.643i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.765 - 0.643i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (1132, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ -0.765 - 0.643i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1014193319 - 0.2783902023i$
$L(\frac12,\chi)$  $\approx$  $0.1014193319 - 0.2783902023i$
$L(\chi,1)$  $\approx$  0.6581040212 + 0.008437261544i
$L(1,\chi)$  $\approx$  0.6581040212 + 0.008437261544i

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

−18.79613257480099790231575118027, −18.125301838593646942487218181439, −17.964706922044315288385261992, −16.61744961147563703837428663111, −15.931065129104070461356250624599, −15.47593680157837664025291852146, −14.768429911585578796700948432814, −14.0676208246253633662182793960, −13.32862479346428136256149147732, −12.315313827676429468812558705769, −11.74011493004332850462595358532, −11.28104924344867588229901735395, −10.36533187698672296244610296555, −9.65558368055595200803427403102, −8.61256703388266812244206938417, −8.455837274599405959902833103361, −7.80227452889145855469655806367, −6.880902517509513304445402177303, −6.297979099526692077379907833050, −5.859850968168734336871671812557, −4.24854325329072045466128937738, −3.25935637102313566682709448291, −2.65106716028933175440987687828, −2.18147514229162515794030078723, −1.08222147042823100348208708764, 0.112724666919900944121248814421, 1.26124035687788925390261260751, 2.08672120471872153289965786928, 3.044481141944500168111959943916, 3.97127742679769342781780730252, 4.40288915954499294016961186793, 5.39607511958577850534476356084, 6.44382920348260692100695062393, 7.347381494076356270472899688630, 7.99296021336836327316192892297, 8.46086515057705102138568878504, 9.05070822155651848170532327434, 9.96966628601594876921356919489, 10.48223799308609384910335713805, 10.90403355336891642020464009043, 11.91338190951349166127617715086, 12.76127770398440621443495547902, 13.36732384905557865175563810287, 14.214927707948074268033920818795, 15.28258816349025078683726477792, 15.63019848686255686578835783885, 15.98940141696511801165424714321, 17.01225271112288582098639252994, 17.31329952524697205211955731306, 18.08750558455971387012043537018

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