Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.115 - 0.993i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.5 − 0.866i)2-s − 3-s + (−0.5 − 0.866i)4-s i·5-s + (−0.5 + 0.866i)6-s + 7-s − 8-s + 9-s + (−0.866 − 0.5i)10-s + (0.866 − 0.5i)11-s + (0.5 + 0.866i)12-s − 13-s + (0.5 − 0.866i)14-s + i·15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s,χ)  = 1  + (0.5 − 0.866i)2-s − 3-s + (−0.5 − 0.866i)4-s i·5-s + (−0.5 + 0.866i)6-s + 7-s − 8-s + 9-s + (−0.866 − 0.5i)10-s + (0.866 − 0.5i)11-s + (0.5 + 0.866i)12-s − 13-s + (0.5 − 0.866i)14-s + i·15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.115 - 0.993i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.115 - 0.993i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.115 - 0.993i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (880, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 4033,\ (0:\ ),\ 0.115 - 0.993i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.224240524 - 1.089632887i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.224240524 - 1.089632887i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8821456079 - 0.6235996929i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8821456079 - 0.6235996929i\)

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

−18.251722106294764563084354122335, −17.75475772557752222310561935195, −17.25250792042351450121808371751, −16.78793319593020543658144487133, −15.742042642078181162803771828129, −15.12910845056242435055691108199, −14.80744669400157911741651367472, −13.85325944380930699585084634553, −13.46319540327960795078028210505, −12.13314083213281463353626956575, −11.9847591404660039014056258177, −11.21978816461441832859514467576, −10.5354825072250011350310051822, −9.517991270268595270343097308234, −8.96145523998419893494440005032, −7.65394561946408750842191343063, −7.16512083871965497596491035127, −6.86892352996480991411680456617, −5.94463321268029372422408361244, −5.13471980856072013562621312091, −4.65350685447511078927314206395, −3.96386107595626811954339508331, −2.83888754204704760947005492600, −2.009833261942064048589162648841, −0.61369512296316966655420268230, 0.81652809050941534452501779496, 1.41660615811100037006739944675, 2.02707874839966807431845672785, 3.435880073138975377194772138423, 4.20014184642216975629420466988, 4.88066061095832234454703603818, 5.25047933328308991575676633382, 6.036179142389657614841550850156, 6.86717230665485641686834393672, 7.9528387414879251130604336699, 8.78378552949522468646724245386, 9.41610730685595831037580289228, 10.27312559981487046816693076751, 10.89999359889112408212628012660, 11.60677784340297737133254397796, 12.09375484575738772039442109567, 12.58342678105092435630059971991, 13.3070132475920723478731717220, 14.14758549057963100587164350262, 14.7880461097615734446376616688, 15.50180119496553363519035265875, 16.52673747141422941814116265522, 17.01587971778355120159945639993, 17.60606273962469422381834841279, 18.27761360051117724347111575109

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