↑

Properties

Label 1-6017-6017.1015-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.997 + 0.0772i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.804 − 0.593i)2-s + (0.473 − 0.880i)3-s + (0.295 − 0.955i)4-s + (0.821 − 0.570i)5-s + (−0.141 − 0.989i)6-s + (0.229 − 0.973i)7-s + (−0.328 − 0.944i)8-s + (−0.550 − 0.834i)9-s + (0.322 − 0.946i)10-s + (−0.700 − 0.713i)12-s + (0.858 + 0.512i)13-s + (−0.393 − 0.919i)14-s + (−0.113 − 0.993i)15-s + (−0.824 − 0.565i)16-s + (0.665 + 0.746i)17-s + (−0.938 − 0.344i)18-s + ⋯
L(s)  = 1  + (0.804 − 0.593i)2-s + (0.473 − 0.880i)3-s + (0.295 − 0.955i)4-s + (0.821 − 0.570i)5-s + (−0.141 − 0.989i)6-s + (0.229 − 0.973i)7-s + (−0.328 − 0.944i)8-s + (−0.550 − 0.834i)9-s + (0.322 − 0.946i)10-s + (−0.700 − 0.713i)12-s + (0.858 + 0.512i)13-s + (−0.393 − 0.919i)14-s + (−0.113 − 0.993i)15-s + (−0.824 − 0.565i)16-s + (0.665 + 0.746i)17-s + (−0.938 − 0.344i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.997 + 0.0772i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1015, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.997 + 0.0772i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1894096564 - 4.895528457i\)
\(L(\frac12)\) \(\approx\) \(-0.1894096564 - 4.895528457i\)
\(L(1)\) \(\approx\) \(1.378478353 - 1.983312671i\)
\(L(1)\) \(\approx\) \(1.378478353 - 1.983312671i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.804 - 0.593i)T \)
3 \( 1 + (0.473 - 0.880i)T \)
5 \( 1 + (0.821 - 0.570i)T \)
7 \( 1 + (0.229 - 0.973i)T \)
13 \( 1 + (0.858 + 0.512i)T \)
17 \( 1 + (0.665 + 0.746i)T \)
19 \( 1 + (0.601 + 0.798i)T \)
23 \( 1 + (0.990 - 0.137i)T \)
29 \( 1 + (0.949 + 0.312i)T \)
31 \( 1 + (0.106 - 0.994i)T \)
37 \( 1 + (0.0930 - 0.995i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.539 + 0.842i)T \)
47 \( 1 + (-0.262 - 0.964i)T \)
53 \( 1 + (0.909 - 0.415i)T \)
59 \( 1 + (0.644 + 0.764i)T \)
61 \( 1 + (-0.933 + 0.357i)T \)
67 \( 1 + (-0.700 - 0.713i)T \)
71 \( 1 + (-0.993 - 0.117i)T \)
73 \( 1 + (-0.980 + 0.198i)T \)
79 \( 1 + (0.634 + 0.773i)T \)
83 \( 1 + (0.958 + 0.285i)T \)
89 \( 1 + (0.0517 - 0.998i)T \)
97 \( 1 + (0.521 + 0.853i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.80168402385244517972104024749, −17.40152489917267030758239166743, −16.44494359415039154473526670119, −15.89330303535343789589078765443, −15.2827686084222183814092000387, −14.91118309209783013732605304881, −14.1412146749320437520637415946, −13.63609104810413119102945285851, −13.1802872929890695187557927635, −12.08631557414484739068582371508, −11.53103249433832201213250412775, −10.77383192329682271003517279226, −10.11568697404051489411341054194, −9.16851592073264490160141069285, −8.791115579725076397920840189795, −8.04288365461161339438602527327, −7.18475602137275692909277897691, −6.42847647968189720641014398914, −5.646960132211041320057532914819, −5.16321205826058646177126709805, −4.67583698795155814420712185208, −3.399230170900993819144670498215, −3.02512985459548047016841462736, −2.548655404036198193266476750354, −1.457897240640437396747065344028, 0.84215103174053331626409078762, 1.350674364565874723061928681146, 1.83361042806282984415295402082, 2.83617772227677330268102570424, 3.57557603479660555710305496668, 4.21485848474823807432048749911, 5.11664028923158221177143993833, 5.881744308133265409639403292916, 6.42135002092475364664775015268, 7.11117196092078467870215294726, 7.99700194270192509433147415437, 8.71769721188487274833871684823, 9.48171538260524457195329365448, 10.18424074919573952476353015233, 10.774405530075417524484840281974, 11.72472745093676604397077487560, 12.160089958720440384243728703731, 13.13279545414850609096859634729, 13.3020517974613611969973498290, 13.85230596729161633753111115713, 14.577920344294565049728294910556, 14.90796508322647279860373691574, 16.235879376534178409529006949909, 16.64211885720143534320180869515, 17.47190715050903353811405094242

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