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Properties

Label 1-6017-6017.255-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: $\chi_{6017} (255, \cdot )$
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

−17.7727494368492548065868126630, −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