Properties

Degree $1$
Conductor $6017$
Sign $0.790 - 0.612i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.998 + 0.0621i)2-s + (0.550 + 0.834i)3-s + (0.992 − 0.123i)4-s + (0.127 + 0.991i)5-s + (−0.601 − 0.798i)6-s + (0.734 − 0.678i)7-s + (−0.982 + 0.185i)8-s + (−0.393 + 0.919i)9-s + (−0.188 − 0.982i)10-s + (0.650 + 0.759i)12-s + (−0.473 − 0.880i)13-s + (−0.691 + 0.722i)14-s + (−0.757 + 0.652i)15-s + (0.969 − 0.246i)16-s + (0.348 − 0.937i)17-s + (0.335 − 0.942i)18-s + ⋯
L(s,χ)  = 1  + (−0.998 + 0.0621i)2-s + (0.550 + 0.834i)3-s + (0.992 − 0.123i)4-s + (0.127 + 0.991i)5-s + (−0.601 − 0.798i)6-s + (0.734 − 0.678i)7-s + (−0.982 + 0.185i)8-s + (−0.393 + 0.919i)9-s + (−0.188 − 0.982i)10-s + (0.650 + 0.759i)12-s + (−0.473 − 0.880i)13-s + (−0.691 + 0.722i)14-s + (−0.757 + 0.652i)15-s + (0.969 − 0.246i)16-s + (0.348 − 0.937i)17-s + (0.335 − 0.942i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.790 - 0.612i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.790 - 0.612i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.790 - 0.612i$
Motivic weight: \(0\)
Character: $\chi_{6017} (255, \cdot )$
Sato-Tate group: $\mu(910)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.790 - 0.612i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(1.044658890 - 0.3573568259i\)
\(L(\frac12,\chi)\) \(\approx\) \(1.044658890 - 0.3573568259i\)
\(L(\chi,1)\) \(\approx\) \(0.8405079870 + 0.1941900676i\)
\(L(1,\chi)\) \(\approx\) \(0.8405079870 + 0.1941900676i\)

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

−17.7727494368492548065868126630, −17.39963427380244407920723863818, −16.52350397113894782943766229603, −16.07167693726029011055610528323, −15.17425271319762458841138184688, −14.45740557111716806929620331418, −14.048493430502151547320077015916, −12.869803937802372479302650106959, −12.40532457718237811908302676514, −11.94945125486511990896271249332, −11.31271254656918028014570135287, −10.3536134460130497707852490778, −9.488275849275105529898934028097, −8.85438695649605714264478899119, −8.66109890585522414214558269174, −7.75937331406013001706883996308, −7.41702558536638118665364310469, −6.3666039341078086477680766304, −5.841495555409495006196869772636, −4.93088780329911050289735667416, −3.99678643365621654621477882956, −2.86734384007629208892392712232, −2.29052309020206436688097674450, −1.33795637721213921117989546780, −1.239534691574973162304780009691, 0.346127879350599951599753991841, 1.55152712653290433204929825937, 2.38952324963288590496162798734, 3.06710584452332585499665425433, 3.607380903455133029572231855320, 4.69675102362820874064628180583, 5.485095268928385199783924970157, 6.20018502823178159574412381266, 7.370616452697235819207292073566, 7.67053341055611433418768884715, 8.049870293718321137481681697660, 9.21131916480343542152020367134, 9.80152401493989094067766029423, 10.12895651754531661992701615471, 10.83219464378266323155898958470, 11.471934508460126908274528494088, 11.88513443208541603337658955792, 13.34385701190680577673380072999, 13.97619550650870956813869138901, 14.47493928752396662130324440032, 15.34604108384700297551396798067, 15.43707609072428732178755777099, 16.43652772061228480605179948686, 17.00615816035626769356673449281, 17.78354005397407594608872516348

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