Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.399 + 0.916i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.173 − 0.984i)2-s + (0.0581 − 0.998i)3-s + (−0.939 − 0.342i)4-s + (−0.802 + 0.597i)5-s + (−0.973 − 0.230i)6-s + (0.396 − 0.918i)7-s + (−0.5 + 0.866i)8-s + (−0.993 − 0.116i)9-s + (0.448 + 0.893i)10-s + (−0.448 + 0.893i)11-s + (−0.396 + 0.918i)12-s + (0.396 − 0.918i)13-s + (−0.835 − 0.549i)14-s + (0.549 + 0.835i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯
L(s,χ)  = 1  + (0.173 − 0.984i)2-s + (0.0581 − 0.998i)3-s + (−0.939 − 0.342i)4-s + (−0.802 + 0.597i)5-s + (−0.973 − 0.230i)6-s + (0.396 − 0.918i)7-s + (−0.5 + 0.866i)8-s + (−0.993 − 0.116i)9-s + (0.448 + 0.893i)10-s + (−0.448 + 0.893i)11-s + (−0.396 + 0.918i)12-s + (0.396 − 0.918i)13-s + (−0.835 − 0.549i)14-s + (0.549 + 0.835i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.399 + 0.916i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.399 + 0.916i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.399 + 0.916i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (2983, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.399 + 0.916i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.4804795171 - 0.3146632793i$
$L(\frac12,\chi)$  $\approx$  $-0.4804795171 - 0.3146632793i$
$L(\chi,1)$  $\approx$  0.4338633176 - 0.6597335641i
$L(1,\chi)$  $\approx$  0.4338633176 - 0.6597335641i

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

−19.01926193321388003442818263507, −18.17758310095311388107470645242, −17.45562529115539544784184019166, −16.71676247573195600993568984827, −16.05332237414291994289277824594, −15.69156430936990616675567189352, −15.265874007931044766865024860374, −14.428527108438952357538036139, −13.79032518456878180111357074026, −13.027517226795983489399125322, −12.18016321743108975516641163535, −11.28932144111343181758568802042, −11.080141818567604163030860719371, −9.613753415913106215912527926671, −9.12380767549878823872652793613, −8.57798652366362511825009160009, −8.08998165377729772062113526294, −7.22779249402054506956196959762, −6.08825605366177145080827706412, −5.5790490790550403064179256527, −4.859809162954704502972083422982, −4.28930991922470349053107964269, −3.53494641268319801598103873693, −2.80034681342950641677452820482, −1.305071566329491278941675393101, 0.196375239833257784579750673836, 0.87652650706569144622741866570, 2.02140775545851558669560415717, 2.54645347232828828637698686183, 3.51056434072972259362777890223, 4.0710743220734746729852970672, 5.00513239945238653294678585198, 5.81283134769770415252489295066, 6.88050589459471885197627973347, 7.52271508129527778535762377562, 7.99964804103502065082797013492, 8.78828072576057311039753983572, 9.859316457172368654187145610698, 10.58893747800139627785713693704, 11.06557974153455967489814328919, 11.7813563864011730768847165071, 12.362848362079861228145493423, 13.08737398856825132959507781495, 13.636560413968864207419428344657, 14.33988891502509950176368720198, 14.91827191946473335657616943355, 15.697839704967882939128705114493, 16.82918514240328814743210272390, 17.6318960439041241740475541208, 18.13317138128076779515369332545

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