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Properties

Label 1-6017-6017.1227-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.790 + 0.612i$
Analytic cond. $27.9428$
Root an. cond. $27.9428$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.790 + 0.612i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.790 + 0.612i)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.998 - 0.0621i)T \)
3 \( 1 + (0.550 - 0.834i)T \)
5 \( 1 + (0.127 - 0.991i)T \)
7 \( 1 + (0.734 + 0.678i)T \)
13 \( 1 + (-0.473 + 0.880i)T \)
17 \( 1 + (0.348 + 0.937i)T \)
19 \( 1 + (0.987 - 0.158i)T \)
23 \( 1 + (-0.322 - 0.946i)T \)
29 \( 1 + (0.0310 + 0.999i)T \)
31 \( 1 + (0.328 + 0.944i)T \)
37 \( 1 + (0.405 - 0.914i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.915 - 0.402i)T \)
47 \( 1 + (-0.906 + 0.421i)T \)
53 \( 1 + (0.229 + 0.973i)T \)
59 \( 1 + (-0.958 + 0.285i)T \)
61 \( 1 + (-0.888 + 0.458i)T \)
67 \( 1 + (-0.650 + 0.759i)T \)
71 \( 1 + (0.113 + 0.993i)T \)
73 \( 1 + (-0.996 + 0.0827i)T \)
79 \( 1 + (-0.503 + 0.863i)T \)
83 \( 1 + (0.995 - 0.0965i)T \)
89 \( 1 + (-0.449 - 0.893i)T \)
97 \( 1 + (0.655 + 0.755i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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