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Properties

Label 1-6017-6017.1444-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.764 + 0.644i$
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  + (−0.607 − 0.794i)2-s + (−0.809 + 0.587i)3-s + (−0.262 + 0.964i)4-s + (−0.527 − 0.849i)5-s + (0.958 + 0.285i)6-s + (0.0241 − 0.999i)7-s + (0.926 − 0.377i)8-s + (0.309 − 0.951i)9-s + (−0.354 + 0.935i)10-s + (−0.354 − 0.935i)12-s + (0.309 − 0.951i)13-s + (−0.809 + 0.587i)14-s + (0.926 + 0.377i)15-s + (−0.861 − 0.506i)16-s + (0.981 − 0.192i)17-s + (−0.943 + 0.331i)18-s + ⋯
L(s)  = 1  + (−0.607 − 0.794i)2-s + (−0.809 + 0.587i)3-s + (−0.262 + 0.964i)4-s + (−0.527 − 0.849i)5-s + (0.958 + 0.285i)6-s + (0.0241 − 0.999i)7-s + (0.926 − 0.377i)8-s + (0.309 − 0.951i)9-s + (−0.354 + 0.935i)10-s + (−0.354 − 0.935i)12-s + (0.309 − 0.951i)13-s + (−0.809 + 0.587i)14-s + (0.926 + 0.377i)15-s + (−0.861 − 0.506i)16-s + (0.981 − 0.192i)17-s + (−0.943 + 0.331i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.764 + 0.644i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1444, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.764 + 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1588319095 - 0.4347631058i\)
\(L(\frac12)\) \(\approx\) \(-0.1588319095 - 0.4347631058i\)
\(L(1)\) \(\approx\) \(0.4594668760 - 0.3050823553i\)
\(L(1)\) \(\approx\) \(0.4594668760 - 0.3050823553i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.607 - 0.794i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.527 - 0.849i)T \)
7 \( 1 + (0.0241 - 0.999i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (0.981 - 0.192i)T \)
19 \( 1 + (0.995 + 0.0965i)T \)
23 \( 1 + (0.885 - 0.464i)T \)
29 \( 1 + (-0.443 - 0.896i)T \)
31 \( 1 + (0.399 + 0.916i)T \)
37 \( 1 + (-0.443 - 0.896i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.354 - 0.935i)T \)
47 \( 1 + (-0.989 + 0.144i)T \)
53 \( 1 + (-0.0724 + 0.997i)T \)
59 \( 1 + (0.995 - 0.0965i)T \)
61 \( 1 + (0.958 + 0.285i)T \)
67 \( 1 + (-0.354 - 0.935i)T \)
71 \( 1 + (0.399 - 0.916i)T \)
73 \( 1 + (0.644 + 0.764i)T \)
79 \( 1 + (-0.998 + 0.0483i)T \)
83 \( 1 + (-0.527 - 0.849i)T \)
89 \( 1 + (0.568 + 0.822i)T \)
97 \( 1 + (-0.906 - 0.421i)T \)
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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

−18.21412076876521541735234942423, −17.54123069744159474403779725446, −16.77647991562158918202931931064, −16.17843589935091632854565745439, −15.695660362791550431796311500427, −14.88741152075740299546753414629, −14.43289022922272915651154765605, −13.61620157538709040977215783010, −12.907556347315759089011895952581, −11.82015252894166405915039396681, −11.56482010144141784591066690826, −10.97323019791072205315044676604, −10.000066598295381952959360990046, −9.56390831041220537217179089400, −8.46853728030845660934958810157, −8.06274143788663557732223421854, −7.10365433237229219951300471485, −6.866606004895217465659011162748, −6.093944284025909860186159192353, −5.39875962802821785294110668298, −4.918404053734913017119612042327, −3.7513415561421126890161208442, −2.78289442059626678851314849622, −1.78987434681754144353023661733, −1.14086857770916675154272681035, 0.221734310012548283823752822750, 0.9513340959275251689378702917, 1.40299858835881268773468544718, 2.99931037569785372084914087623, 3.58324627304996879831867084988, 4.11913859498342210926957990267, 5.04065168311760887774678020488, 5.36748802511512052018004910248, 6.662957161780957128323987876566, 7.42439867925970851514908648450, 8.004737113222688288189340114378, 8.74375680578822575696418660213, 9.57500880528405666236591698468, 10.0598619903357771035128476097, 10.68488300094607790724681231019, 11.28846358379821187037988415105, 11.937088270009880680233003791466, 12.46571405714912503573694202163, 13.127484670302263851294112131273, 13.77793540618314305376304715178, 14.82299832285562619314781506928, 15.72933990272223719617730304622, 16.23098060194540211378539136187, 16.75159412896497276528868884762, 17.28926106533463128357861078510

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