Properties

Degree $1$
Conductor $4033$
Sign $-0.705 + 0.709i$
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 + (−0.286 − 0.957i)3-s + (−0.939 + 0.342i)4-s + (0.835 + 0.549i)5-s + (0.893 − 0.448i)6-s + (−0.0581 + 0.998i)7-s + (−0.5 − 0.866i)8-s + (−0.835 + 0.549i)9-s + (−0.396 + 0.918i)10-s + (−0.396 − 0.918i)11-s + (0.597 + 0.802i)12-s + (0.893 − 0.448i)13-s + (−0.993 + 0.116i)14-s + (0.286 − 0.957i)15-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s,χ)  = 1  + (0.173 + 0.984i)2-s + (−0.286 − 0.957i)3-s + (−0.939 + 0.342i)4-s + (0.835 + 0.549i)5-s + (0.893 − 0.448i)6-s + (−0.0581 + 0.998i)7-s + (−0.5 − 0.866i)8-s + (−0.835 + 0.549i)9-s + (−0.396 + 0.918i)10-s + (−0.396 − 0.918i)11-s + (0.597 + 0.802i)12-s + (0.893 − 0.448i)13-s + (−0.993 + 0.116i)14-s + (0.286 − 0.957i)15-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.705 + 0.709i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.705 + 0.709i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.705 + 0.709i$
Motivic weight: \(0\)
Character: $\chi_{4033} (28, \cdot )$
Sato-Tate group: $\mu(54)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.705 + 0.709i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.5928037875 + 1.425347605i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.5928037875 + 1.425347605i\)
\(L(\chi,1)\) \(\approx\) \(0.9347394172 + 0.5319487820i\)
\(L(1,\chi)\) \(\approx\) \(0.9347394172 + 0.5319487820i\)

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

−18.198494440940929979447020007852, −17.509516717648814852729978461315, −16.98797513088367901720615947711, −16.24732632844853220352467661788, −15.61216426879590914877388389859, −14.48735437993488061674341277238, −14.0487022994663427422948056992, −13.44233168313730729973794422093, −12.70867982880517906980973010976, −11.96986680008248032398568913519, −11.16992068764924071120502402067, −10.59834988226879928783491799768, −9.9760545927174608866773321993, −9.4099106712765766295390991017, −8.99054900642531688036241203840, −7.923278929258317963725114992, −6.88396485448207961035452308581, −5.796418337152458225584152154172, −5.31668137546403690158286261356, −4.38831814669187477228834287175, −4.1749419071094377509513482916, −3.11033005824194283973949239353, −2.31973534224755559060733838252, −1.29772058590229779124062603580, −0.48434712199776618305472267636, 1.07998788693745625132498377551, 1.89518512184967207218216607304, 3.162316231982612922127084484097, 3.35181027709759884406210292034, 5.12045195786816212482338082143, 5.59702478010261875414334955479, 6.02470877209145001657945831790, 6.56034284853879895148955544453, 7.57109248619344152636325336426, 8.0246830319709186681026060625, 8.93026803599939317644036623774, 9.34794092854373263823637299294, 10.579326652954156040266499625051, 11.148869701639715231728640317001, 12.08997122070677494318607472686, 12.9249651753774753348702230870, 13.25982700229466764004881490356, 13.94350506593989630246687761261, 14.580612083438497127144302602670, 15.2725341730932199865139954819, 16.107353731528118114552590791407, 16.648785120514970448342066548067, 17.528882550001236990317777696228, 18.14643766711339010486866312890, 18.340849595110221018880043116552

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