Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.485 + 0.874i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.984 − 0.173i)2-s + (−0.835 − 0.549i)3-s + (0.939 + 0.342i)4-s + (0.993 − 0.116i)5-s + (0.727 + 0.686i)6-s + (0.597 + 0.802i)7-s + (−0.866 − 0.5i)8-s + (0.396 + 0.918i)9-s + (−0.998 − 0.0581i)10-s + (−0.998 + 0.0581i)11-s + (−0.597 − 0.802i)12-s + (−0.802 + 0.597i)13-s + (−0.448 − 0.893i)14-s + (−0.893 − 0.448i)15-s + (0.766 + 0.642i)16-s + (0.642 − 0.766i)17-s + ⋯
L(s,χ)  = 1  + (−0.984 − 0.173i)2-s + (−0.835 − 0.549i)3-s + (0.939 + 0.342i)4-s + (0.993 − 0.116i)5-s + (0.727 + 0.686i)6-s + (0.597 + 0.802i)7-s + (−0.866 − 0.5i)8-s + (0.396 + 0.918i)9-s + (−0.998 − 0.0581i)10-s + (−0.998 + 0.0581i)11-s + (−0.597 − 0.802i)12-s + (−0.802 + 0.597i)13-s + (−0.448 − 0.893i)14-s + (−0.893 − 0.448i)15-s + (0.766 + 0.642i)16-s + (0.642 − 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.485 + 0.874i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.485 + 0.874i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.485 + 0.874i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (3283, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (1:\ ),\ -0.485 + 0.874i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4024886344 + 0.6841127226i$
$L(\frac12,\chi)$  $\approx$  $0.4024886344 + 0.6841127226i$
$L(\chi,1)$  $\approx$  0.6307259067 + 0.01284102874i
$L(1,\chi)$  $\approx$  0.6307259067 + 0.01284102874i

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

−17.59177674886228224531990389339, −17.55176846035225618640577902508, −16.9188701617357614369815132772, −16.396753177053479907038063021488, −15.32693272182311955630193878265, −14.98742118423655579453720339151, −14.23556761355955407373091399513, −13.077256865557489641554598856152, −12.63471953710493434805433401926, −11.50731165479651720780141688209, −10.797213209594891283529218274035, −10.51835564730424302890632983591, −9.9132536531620740717201126988, −9.25767941822101271739321504724, −8.36208032614279300189411594103, −7.49208560562938512026709167717, −6.94153045824868422168748468759, −6.09397048726300090252153058349, −5.2913250945986186653348090558, −5.00988178023888851794847834405, −3.707414547874498976065090556545, −2.67914043464865273182131239079, −1.86019069888545647571114984941, −0.88418182462430013827518108779, −0.23325505488699915886291775120, 0.91459782332499266604822041487, 1.66957363987751293932318837727, 2.34107160561610903758220841008, 2.87427485938495016100965130316, 4.62052441413868324607449140226, 5.2976374458458233030939195564, 5.878916974953654703535059868701, 6.64614837966483813306602797561, 7.407398929146515146272094452362, 8.03468957164298950783813482106, 8.83822084452055830171937948617, 9.63585491712006138171631597951, 10.20742534226521315999782809772, 10.909771630636248472594391283432, 11.58684861905815513088761128035, 12.287208704973647817665193549917, 12.75142227639302989766058148260, 13.57464408687952145890813108634, 14.584007483729359692300683855194, 15.17828463301893832217968164621, 16.32973884808925612688895534392, 16.57500613233339128651669032226, 17.329890988126803280692461384865, 17.97251100598868529881373900487, 18.33190442837475349824319071491

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