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 + (0.5 + 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.399 - 0.916i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (468, \cdot )$
Sato-Tate  :  $\mu(36)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 4033,\ (0:\ ),\ -0.399 - 0.916i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(-1.201797488 + 1.835264155i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(-1.201797488 + 1.835264155i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5224647585 + 1.263253291i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5224647585 + 1.263253291i\)

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

−18.28497635419137732776396499429, −17.35932379810735542786466104226, −16.88791883120586303164189239931, −16.258040077339127622410999716696, −14.79469626381313861572465593397, −14.168662677873432217148066320926, −13.781538510611355467582839964009, −13.26219181899544398711631992606, −12.62139079662982840606183064256, −11.94964097299602749172113629846, −11.25742250978221740526006794442, −10.37176309903003677494607693151, −9.56266873682250095169517537834, −9.17490557824302220302581259765, −8.50835553582425918839893747595, −7.5092168956122228020322280049, −6.73844078655530567463333556290, −5.953400616021632709615023821393, −5.20001071631421679782513995339, −4.25953394674629224746323217417, −3.39564842044323775617621340183, −2.781886827426288945993854424546, −1.9542111626870687550386063074, −1.078124126625894864363243779242, −0.59052500562118616385907973905, 1.49023316155127231843778915927, 2.65429163070816985479080416526, 3.208815904326198907366157324579, 3.948102240300652018852512847560, 5.15022654717960607819389858679, 5.40754878567536305404730921287, 6.1022692371056874543017333552, 7.05963664978411861954773546516, 7.75221485506351123312240752177, 8.493003763738392861567977771906, 9.44036829020558988200717921383, 9.76825607935126591024793592309, 10.117674604444591783506768595651, 11.34770079456904315701776308368, 12.42790040664121210083525119473, 12.86182639653228288861364052961, 13.80444624016815472607358004813, 14.41727668927221766815468069893, 14.85812077679078336414958043615, 15.51048460154471408896597696597, 15.94022636868659483158184931108, 17.02089102141492817607424997721, 17.28004634503356495104027330503, 18.09736310771417188815355259317, 18.85866358288110188153873809565

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