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Properties

Label 1-6017-6017.2996-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: $\chi_{6017} (2996, \cdot )$
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

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

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