Properties

Degree $1$
Conductor $4033$
Sign $0.368 + 0.929i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.368 + 0.929i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.368 + 0.929i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.368 + 0.929i$
Motivic weight: \(0\)
Character: $\chi_{4033} (5, \cdot )$
Sato-Tate group: $\mu(108)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.368 + 0.929i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(1.428837878 + 0.9707201182i\)
\(L(\frac12,\chi)\) \(\approx\) \(1.428837878 + 0.9707201182i\)
\(L(\chi,1)\) \(\approx\) \(0.9853673681 + 0.03016885842i\)
\(L(1,\chi)\) \(\approx\) \(0.9853673681 + 0.03016885842i\)

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

−18.197644399283377642040588499480, −17.50321275271553340316762067238, −16.84561687133826799520860900617, −16.13994569762699864944668049296, −15.381958963293567955546802326225, −15.00403192976680607640862966528, −13.83746479083039909029832833047, −13.4011621865325036035502936463, −12.78672202130694146722954877156, −12.56225321978340925777090397859, −11.56259406896204447324224805656, −10.18301614302073885949442502987, −9.64578277154559468706202095161, −8.88614468301939917863470851347, −8.36649839477934255574813934758, −7.61103334034142185589043821154, −7.070851912705298162503455242933, −6.0813098927908197456972779606, −5.47853727677346004717733115547, −5.254350877229853864030425483959, −3.80889715843417021365610048900, −3.09047320559329897494539149948, −2.10816121439839899341584081894, −1.029433594567479211453163739487, −0.38154436026765438902012639128, 0.669595600590436342061295637023, 1.911857954459251331738790415040, 2.74364902382921215642617425306, 3.37702039574525565919793913352, 3.731418851784030285542550491253, 4.77215161006818682892677282176, 5.56216583325181245674099883641, 6.23896601574538577451734972328, 7.40300604738715089097850574008, 8.10251620471289315113951512628, 9.20238229582177031352794813431, 9.55051050304699163909659156449, 10.19288730845898794551047218019, 10.87257998726100658687494925923, 11.119396128270198192950497110413, 12.18506791034863615492133339506, 13.076228224566953234893003814472, 13.70264544361920014865712844306, 14.27667905766171491730172927269, 14.68876735610876192304478809925, 15.80836695408209312712471199403, 16.538618925112615852047475467732, 16.84630984460563177123773984637, 18.09997291695290271357003648754, 18.586369881702596357002774676235

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