Properties

Degree $1$
Conductor $4033$
Sign $0.569 - 0.821i$
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 + (−0.893 + 0.448i)3-s + (−0.5 + 0.866i)4-s + (0.116 + 0.993i)5-s + (0.835 + 0.549i)6-s + (0.597 + 0.802i)7-s + 8-s + (0.597 − 0.802i)9-s + (0.802 − 0.597i)10-s + (−0.802 − 0.597i)11-s + (0.0581 − 0.998i)12-s + (−0.993 + 0.116i)13-s + (0.396 − 0.918i)14-s + (−0.549 − 0.835i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s,χ)  = 1  + (−0.5 − 0.866i)2-s + (−0.893 + 0.448i)3-s + (−0.5 + 0.866i)4-s + (0.116 + 0.993i)5-s + (0.835 + 0.549i)6-s + (0.597 + 0.802i)7-s + 8-s + (0.597 − 0.802i)9-s + (0.802 − 0.597i)10-s + (−0.802 − 0.597i)11-s + (0.0581 − 0.998i)12-s + (−0.993 + 0.116i)13-s + (0.396 − 0.918i)14-s + (−0.549 − 0.835i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.569 - 0.821i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.569 - 0.821i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.569 - 0.821i$
Motivic weight: \(0\)
Character: $\chi_{4033} (3103, \cdot )$
Sato-Tate group: $\mu(108)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.569 - 0.821i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.3648718529 - 0.1909793808i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.3648718529 - 0.1909793808i\)
\(L(\chi,1)\) \(\approx\) \(0.5194097889 + 0.007227341954i\)
\(L(1,\chi)\) \(\approx\) \(0.5194097889 + 0.007227341954i\)

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

−18.32573464547741338216310494190, −17.622817370852962947419997954697, −17.0705298251858989750583529398, −16.864769667732214951233328443946, −16.12990570861607774725022964591, −15.3063859128259349256874250696, −14.6027477304689545958766359783, −13.82252745231106198567694429953, −12.973554650270838453812312801187, −12.63089664672317835063283998214, −11.77246034936774229201711792238, −10.71524759318338357690919084333, −10.253928555483850195742357697039, −9.71883863418564222412489461752, −8.55306847303956453161735633844, −7.83290040411293789535797826665, −7.63142572676399290407353374773, −6.63578365987554927169702763126, −5.91053682150676142859857824129, −5.19876568868355547241760511419, −4.61923836000071835689965687211, −4.126119968381935892332711285122, −2.10336092424419424204696849402, −1.59943006377576012149465259059, −0.60712647938325583781366571504, 0.25288238444924502522302443988, 1.608771632677088923099218605114, 2.5225130128328593157684376155, 2.98565631512921142725393363691, 4.06006977930075814511990160709, 4.81326471963073866170268058010, 5.56255866921269115367384113773, 6.2696732812216256459811721070, 7.45140029260981040528963491532, 7.75215505262701280742510840187, 8.94950688514263477050396443628, 9.54724447544761538685237835817, 10.23447684014210579785257297470, 10.8968950858126636023191565766, 11.26424358485446487586895448556, 12.13824343898697996807350911812, 12.35960391390640357030190985543, 13.598447700860302840094858408132, 14.147928439069278967995766065756, 15.21868090402941295129192008586, 15.56898050154846911657695226386, 16.568840659970892338708859834264, 17.20131862523450240132673085851, 17.93268873641747162141087862101, 18.20344820217715324680080177515

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