Properties

Degree $1$
Conductor $4033$
Sign $-0.792 + 0.609i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 + 0.866i)2-s + (−0.686 − 0.727i)3-s + (−0.5 − 0.866i)4-s + (−0.835 + 0.549i)5-s + (0.973 − 0.230i)6-s + (−0.0581 − 0.998i)7-s + 8-s + (−0.0581 + 0.998i)9-s + (−0.0581 − 0.998i)10-s + (−0.0581 + 0.998i)11-s + (−0.286 + 0.957i)12-s + (−0.835 + 0.549i)13-s + (0.893 + 0.448i)14-s + (0.973 + 0.230i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s,χ)  = 1  + (−0.5 + 0.866i)2-s + (−0.686 − 0.727i)3-s + (−0.5 − 0.866i)4-s + (−0.835 + 0.549i)5-s + (0.973 − 0.230i)6-s + (−0.0581 − 0.998i)7-s + 8-s + (−0.0581 + 0.998i)9-s + (−0.0581 − 0.998i)10-s + (−0.0581 + 0.998i)11-s + (−0.286 + 0.957i)12-s + (−0.835 + 0.549i)13-s + (0.893 + 0.448i)14-s + (0.973 + 0.230i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.792 + 0.609i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.792 + 0.609i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.792 + 0.609i$
Motivic weight: \(0\)
Character: $\chi_{4033} (9, \cdot )$
Sato-Tate group: $\mu(27)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.792 + 0.609i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.05522457301 + 0.1623661206i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.05522457301 + 0.1623661206i\)
\(L(\chi,1)\) \(\approx\) \(0.4458228864 + 0.07111418573i\)
\(L(1,\chi)\) \(\approx\) \(0.4458228864 + 0.07111418573i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.214767218147603424180755612094, −17.564741489355541022961584974006, −16.83271919478099485776201434604, −16.2512469305592407595393420804, −15.74345588460138783712199986118, −14.98260087123499456473166635138, −14.151216160465785431359369933623, −12.95635937003324900125829907279, −12.349296684048328141304203734653, −12.03519192835188942967490126243, −11.32360528478853709243331444603, −10.66027136738943124584936154753, −10.021704355909842647970476561882, −9.23111685106493093350616640263, −8.52523556629622400437283608549, −8.16768027190524676629727663828, −7.13489246615303814140255195640, −5.82862819698783466943792125380, −5.49260922882445800252977298504, −4.389596105650043300891319616883, −3.93892686614800296536835384478, −3.124283752573616176302183126731, −2.268423998391343824590420296963, −1.07027349551373244747815134927, −0.10914339197986095606115498987, 0.69670294507938539684390450382, 1.791449349986978962428792226751, 2.69666423717193740323106525479, 4.18637692013394109968943564878, 4.590470806206444692526775491031, 5.359794388739116779726527229281, 6.542908365544654829021825895572, 6.94550925170350435182414151264, 7.42019414274255704795594962728, 7.808682985251204359189978287699, 8.933247094134058383698169682238, 9.78236659126402206447563443774, 10.48499592534698931115485785390, 11.08439028685505375385333614042, 11.82414763808043081630164353626, 12.53802282790248310944631176489, 13.49047202485933119868445002619, 14.029349686220094533749358947798, 14.67465572690873176849923569766, 15.558690115275049335290825380746, 16.10314397375093191052217189648, 16.752607581418735791081569074770, 17.44508752851650946502230563532, 18.03197609963613156175509414085, 18.38694297269793132774822640504

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