Properties

Degree $1$
Conductor $4033$
Sign $0.206 + 0.978i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.206 + 0.978i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.206 + 0.978i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.206 + 0.978i$
Motivic weight: \(0\)
Character: $\chi_{4033} (3876, \cdot )$
Sato-Tate group: $\mu(54)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.206 + 0.978i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(1.256394534 + 1.018878112i\)
\(L(\frac12,\chi)\) \(\approx\) \(1.256394534 + 1.018878112i\)
\(L(\chi,1)\) \(\approx\) \(0.9000706798 + 0.4958228321i\)
\(L(1,\chi)\) \(\approx\) \(0.9000706798 + 0.4958228321i\)

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

−18.352338012813722241845828009988, −17.89354564582629283137986229240, −17.32699780090007162765067354259, −16.52710458951437633910750770199, −15.450933439506255334957177612337, −14.46211745851343613670549913235, −14.096947418294754043487831717048, −13.75646868905948975662086367273, −12.37995497032525072871169125306, −12.10071737237713394434478421128, −11.51679468511607196483645250828, −11.016917213692122536346734825474, −10.257216171834575723199418001630, −9.50101906137391263877597498718, −8.67199265544853770097212235081, −7.807444290885602611470906979489, −6.96492844721699068923487679559, −6.42340192607156645020430259597, −5.33425860921760467686328916325, −4.95437552601455933220113344393, −4.00634589185433263416288719090, −2.89417930731908470082890962858, −2.35417085715722736576745009747, −1.58085720134223525950517334197, −0.71142892600351486204383723216, 0.805116545687943772997297907545, 1.397261232007934223683826535680, 3.15301076281563305887682598752, 4.140274390726486995639413464733, 4.475992017575458480528282391098, 5.17355622473234140863949254645, 5.789068583547586543289246212425, 6.46452950125631049764140587953, 7.60171565959219193450959609422, 7.95436830379412853817210978574, 8.95070335366432042178854228163, 9.54283899930107450782839359615, 10.05127629189271529449276678054, 11.14283381117019964294568372358, 11.90593142628423662446256379636, 12.40251067810628941960670795076, 13.20716481875228029292080813495, 14.117402496712627513800442992778, 14.61394432590360896268294889941, 15.36799632137898616667453746645, 15.89095434254397023446275499350, 16.76467994774582482839602507038, 17.14625338668375395868125285226, 17.52130876309622381904162015945, 18.08593712996281882195900895545

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