Properties

Degree $1$
Conductor $4033$
Sign $-0.999 + 0.0398i$
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 + (−0.597 − 0.802i)3-s + (−0.5 + 0.866i)4-s + (0.230 − 0.973i)5-s + (−0.396 + 0.918i)6-s + (−0.286 − 0.957i)7-s + 8-s + (−0.286 + 0.957i)9-s + (−0.957 + 0.286i)10-s + (0.957 + 0.286i)11-s + (0.993 − 0.116i)12-s + (0.973 + 0.230i)13-s + (−0.686 + 0.727i)14-s + (−0.918 + 0.396i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s,χ)  = 1  + (−0.5 − 0.866i)2-s + (−0.597 − 0.802i)3-s + (−0.5 + 0.866i)4-s + (0.230 − 0.973i)5-s + (−0.396 + 0.918i)6-s + (−0.286 − 0.957i)7-s + 8-s + (−0.286 + 0.957i)9-s + (−0.957 + 0.286i)10-s + (0.957 + 0.286i)11-s + (0.993 − 0.116i)12-s + (0.973 + 0.230i)13-s + (−0.686 + 0.727i)14-s + (−0.918 + 0.396i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.999 + 0.0398i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.999 + 0.0398i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.999 + 0.0398i$
Motivic weight: \(0\)
Character: $\chi_{4033} (1720, \cdot )$
Sato-Tate group: $\mu(108)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.999 + 0.0398i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.02475554490 - 1.241789390i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.02475554490 - 1.241789390i\)
\(L(\chi,1)\) \(\approx\) \(0.4893982045 - 0.6416919961i\)
\(L(1,\chi)\) \(\approx\) \(0.4893982045 - 0.6416919961i\)

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

−18.756218232460606749161884223471, −17.943719501520437521271932377056, −17.44005868982453375303635865857, −16.787439379391475685282581101190, −16.02045520657180789876011280815, −15.36863110817951238166422759712, −15.152266455966432057430985521250, −14.16413619831785204314166215437, −13.8147872820873370144450319945, −12.57680622081032097700588168007, −11.659706804059070896166509558984, −11.17340755739595826554618554131, −10.33912989966209042146308653974, −9.721917067021015931879562557921, −9.28251419273081966263313361155, −8.39414515085157261356856087099, −7.70294396225930704833170860757, −6.51803721084686800791356333468, −6.085338604237328558277180446166, −5.83477169932191659350801711315, −4.870570916757049124784322332354, −3.69554093519060816504009121884, −3.38161402126170337501405298182, −1.949104529226246684669289848610, −0.96222674729561064179823277892, 0.665758246754776993751413444449, 1.06365346125995270332051008426, 1.71646540569873675411012860459, 2.833130918547512681814833407842, 3.7441998673217355605585900603, 4.61241045638497408077504471669, 5.128137563265409832888875284728, 6.36388548045940140188697449174, 6.94492507340227415147974298657, 7.6803229410456534459598867586, 8.56452514028214962092429936366, 9.03944363640602541197485295023, 9.96439421222948216314589074414, 10.5268514751523472527854453832, 11.401888789366866000183076604843, 11.987939042938070542295798949851, 12.4221589855759084776745741507, 13.2871897381352549164746591771, 13.73533114185941207358427560036, 14.18142943398600841021046234917, 15.917672382157814347210832817, 16.47652121666211673877230520395, 16.81514629474191852402394710625, 17.62919011617950692459824819406, 18.01798091393276557874117731533

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