Properties

Degree $1$
Conductor $6001$
Sign $-0.634 - 0.772i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + i·11-s − 12-s i·13-s + 14-s + 15-s + 16-s + ⋯
L(s,χ)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + i·11-s − 12-s i·13-s + 14-s + 15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.634 - 0.772i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.634 - 0.772i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.634 - 0.772i$
Motivic weight: \(0\)
Character: $\chi_{6001} (395, \cdot )$
Sato-Tate group: $\mu(4)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.634 - 0.772i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.1465138952 - 0.3100755137i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.1465138952 - 0.3100755137i\)
\(L(\chi,1)\) \(\approx\) \(0.3841619160 - 0.06115100233i\)
\(L(1,\chi)\) \(\approx\) \(0.3841619160 - 0.06115100233i\)

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

−18.1259586880444827901511118086, −17.143822280365947410911988238443, −16.48510616163114829292291990011, −16.19865176170716029597053754699, −15.83460209557686447956547036538, −15.01486314713030691123437407595, −14.07979742309161423771702136875, −13.05526192696596689997171427765, −12.46240185895298657562775517077, −11.72230907253641366420696993827, −11.28001072281329956962672011679, −10.88209279646687110180763648179, −9.89421803852736992240591084999, −9.37900662012277820443846761923, −8.73389349815161739273649489415, −7.739314124141200072128446628286, −7.22549434058403193714668860124, −6.64406193047284268829415538353, −5.93932431849730986700706760826, −5.28410143704459913293371460920, −4.11999066449289336323171567477, −3.466115131081746407709854784513, −2.76996807629407810348724717053, −1.41683340631964138809949499305, −0.82861995289648409014683235974, 0.275458911610593040608095838789, 0.75075537473408059908378025792, 1.94290435599281951284494526741, 2.90624415857761881538121804234, 3.65123298521947063656024521800, 4.49877432209585742739742818903, 5.43233274187182782275548163509, 6.13179172777124695792165178473, 6.87248445534565468717633874894, 7.413150811522188976254797442489, 7.89312200522691258794759656470, 8.880137969015151625129891126425, 9.756963547896428450697372125586, 10.056780006685595705528196206599, 10.86586355276297716548453911457, 11.39596459829866594857540286552, 12.19728941538896521078954507595, 12.55246811048839042860653108801, 13.11165620491389209714099899460, 14.55972693545116648719816648455, 15.27344906332730964408314499020, 15.77582539839862509054647222611, 16.18758500888973924712981930559, 16.846360672013885372360191230713, 17.46519527950509094250172566587

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