Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.603 - 0.797i$
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 + (0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s + (0.866 + 0.5i)5-s + (0.766 − 0.642i)6-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.342 − 0.939i)10-s + (−0.342 − 0.939i)11-s + (−0.766 − 0.642i)12-s + (0.5 + 0.866i)13-s + (0.939 + 0.342i)14-s + i·15-s + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + ⋯
L(s,χ)  = 1  + (−0.173 − 0.984i)2-s + (0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s + (0.866 + 0.5i)5-s + (0.766 − 0.642i)6-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.342 − 0.939i)10-s + (−0.342 − 0.939i)11-s + (−0.766 − 0.642i)12-s + (0.5 + 0.866i)13-s + (0.939 + 0.342i)14-s + i·15-s + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.603 - 0.797i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.603 - 0.797i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.603 - 0.797i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (949, \cdot )$
Sato-Tate  :  $\mu(36)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ -0.603 - 0.797i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2013537067 - 0.4052207164i$
$L(\frac12,\chi)$  $\approx$  $0.2013537067 - 0.4052207164i$
$L(\chi,1)$  $\approx$  0.8957813718 - 0.03618202548i
$L(1,\chi)$  $\approx$  0.8957813718 - 0.03618202548i

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

−18.313339082440784446295822073949, −17.9700016823532055730088499842, −17.39104556702738031827003257050, −16.853235635406039121571802241378, −16.03895485197104804934117852611, −15.219231829474027232966988847968, −14.650044049923908496365122456487, −13.841193925799107062228896523980, −13.22128253590552244765739596051, −12.98503202652883827086968039188, −12.361001631127710154572313583065, −10.88647315501308445410259395157, −10.1756308638187720397106974221, −9.48344495214128318128762881148, −8.94913631458939268679779318374, −7.98810595549817378035284645004, −7.64935653761697596669388890872, −6.77586033822800753722928357167, −6.12475331637046849477946999663, −5.673952499580183509400579496656, −4.50604369927425494045813024364, −3.90685101206915137607367672732, −2.77310094198858929720901519001, −1.717451114725643569323400750007, −1.10023474329357468595219593421, 0.1228049196503547515775112102, 1.788045184033002344517691408994, 2.506085898755237495844933015667, 2.79864137898154963693938264205, 3.78801095874701973224044801371, 4.455832395710024631970731479263, 5.40750448399699257599594609912, 6.02181649095676291512980342034, 6.94700313609560333179260645722, 8.29510497501093847358640926644, 8.884405825394257624411031630588, 9.14522189187012009920319512155, 9.96844387330227647756881965118, 10.6741490073365136572438643219, 11.09084307043381195119391876969, 11.85085869016589265675745888334, 12.894202101932829788694174443578, 13.511992505450505210243691383790, 13.95959173229690523335677228777, 14.66468952857164035024545637481, 15.53802656259148048280805697629, 16.18104978731701742047818294560, 16.93252608562037799144193182813, 17.68269960083775289814719030846, 18.52862901809520414700059897491

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