Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.659 - 0.751i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.766 + 0.642i)2-s + (0.597 − 0.802i)3-s + (0.173 − 0.984i)4-s + (0.686 + 0.727i)5-s + (0.0581 + 0.998i)6-s + (0.973 − 0.230i)7-s + (0.5 + 0.866i)8-s + (−0.286 − 0.957i)9-s + (−0.993 − 0.116i)10-s + (−0.993 + 0.116i)11-s + (−0.686 − 0.727i)12-s + (−0.973 + 0.230i)13-s + (−0.597 + 0.802i)14-s + (0.993 − 0.116i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s,χ)  = 1  + (−0.766 + 0.642i)2-s + (0.597 − 0.802i)3-s + (0.173 − 0.984i)4-s + (0.686 + 0.727i)5-s + (0.0581 + 0.998i)6-s + (0.973 − 0.230i)7-s + (0.5 + 0.866i)8-s + (−0.286 − 0.957i)9-s + (−0.993 − 0.116i)10-s + (−0.993 + 0.116i)11-s + (−0.686 − 0.727i)12-s + (−0.973 + 0.230i)13-s + (−0.597 + 0.802i)14-s + (0.993 − 0.116i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.659 - 0.751i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.659 - 0.751i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.659 - 0.751i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (1248, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.659 - 0.751i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.414904668 - 0.6408248339i$
$L(\frac12,\chi)$  $\approx$  $1.414904668 - 0.6408248339i$
$L(\chi,1)$  $\approx$  1.023092318 - 0.03537233947i
$L(1,\chi)$  $\approx$  1.023092318 - 0.03537233947i

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

−18.580076419165040448054539357, −17.80025843360364227104009896253, −17.2980509656955517363789519513, −16.595877302565640632339880036073, −16.082595385258966551364558871676, −15.14362446573554363893644595590, −14.58954193109601047949751331389, −13.66973881117724363031032025427, −13.03158833833822912905510201073, −12.397922738237946318532360924901, −11.475658870435153187274757539519, −10.8013383353510917501498151019, −10.15475282686133000684254434830, −9.60182877700519905389873842781, −8.927524789398373708287367393579, −8.27092816595327566871081399800, −7.85136916312674949090214672475, −6.923987263250479097856159346764, −5.42856205837010335874054718622, −4.97787529752728638286251814589, −4.420837896994647032769151693425, −3.12531454574648113977572637051, −2.65225349756694207389402282184, −1.880883415723847171436779587786, −1.00784024198874717073031615400, 0.53909761850134728953747674911, 1.73022940620149270390910998582, 2.127928924357946763845186644086, 2.865978169599534906160895027326, 4.16054773642596171408428261868, 5.3160754135368070270617154688, 5.739263638895781666983504433084, 6.78252388052632146734066981570, 7.20638582949505475278644003017, 7.93969664325344592489324024526, 8.34473162956043389917921687016, 9.27440142146155085502867242394, 10.141422375657147621575279824369, 10.41908846239986742272185582242, 11.42280467387576842589572561899, 12.15602755336536599470567950844, 13.24716252099284130304474321812, 13.709026304879001188915682930892, 14.63967626414383231998532912268, 14.77024333315797672074897987211, 15.35146330658624457810868725871, 16.71906799815700533927717853040, 17.1949139618087195679092403398, 17.67278760798809921726797164628, 18.45674154748237838083231876879

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