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Properties

Label 1-6017-6017.2425-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.343 + 0.939i$
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  + (0.215 − 0.976i)2-s + (0.309 + 0.951i)3-s + (−0.906 − 0.421i)4-s + (−0.943 + 0.331i)5-s + (0.995 − 0.0965i)6-s + (−0.861 + 0.506i)7-s + (−0.607 + 0.794i)8-s + (−0.809 + 0.587i)9-s + (0.120 + 0.992i)10-s + (0.120 − 0.992i)12-s + (−0.809 + 0.587i)13-s + (0.309 + 0.951i)14-s + (−0.607 − 0.794i)15-s + (0.644 + 0.764i)16-s + (−0.443 − 0.896i)17-s + (0.399 + 0.916i)18-s + ⋯
L(s)  = 1  + (0.215 − 0.976i)2-s + (0.309 + 0.951i)3-s + (−0.906 − 0.421i)4-s + (−0.943 + 0.331i)5-s + (0.995 − 0.0965i)6-s + (−0.861 + 0.506i)7-s + (−0.607 + 0.794i)8-s + (−0.809 + 0.587i)9-s + (0.120 + 0.992i)10-s + (0.120 − 0.992i)12-s + (−0.809 + 0.587i)13-s + (0.309 + 0.951i)14-s + (−0.607 − 0.794i)15-s + (0.644 + 0.764i)16-s + (−0.443 − 0.896i)17-s + (0.399 + 0.916i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.343 + 0.939i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (2425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.343 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5349964453 + 0.3738176871i\)
\(L(\frac12)\) \(\approx\) \(0.5349964453 + 0.3738176871i\)
\(L(1)\) \(\approx\) \(0.7348074987 - 0.04741663895i\)
\(L(1)\) \(\approx\) \(0.7348074987 - 0.04741663895i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.215 - 0.976i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.943 + 0.331i)T \)
7 \( 1 + (-0.861 + 0.506i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (-0.443 - 0.896i)T \)
19 \( 1 + (-0.527 - 0.849i)T \)
23 \( 1 + (-0.354 - 0.935i)T \)
29 \( 1 + (0.779 + 0.626i)T \)
31 \( 1 + (0.926 - 0.377i)T \)
37 \( 1 + (0.779 + 0.626i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.120 - 0.992i)T \)
47 \( 1 + (-0.998 - 0.0483i)T \)
53 \( 1 + (0.0241 + 0.999i)T \)
59 \( 1 + (-0.527 + 0.849i)T \)
61 \( 1 + (0.995 - 0.0965i)T \)
67 \( 1 + (0.120 - 0.992i)T \)
71 \( 1 + (0.926 + 0.377i)T \)
73 \( 1 + (0.958 - 0.285i)T \)
79 \( 1 + (0.485 + 0.873i)T \)
83 \( 1 + (-0.943 + 0.331i)T \)
89 \( 1 + (-0.748 - 0.663i)T \)
97 \( 1 + (-0.989 + 0.144i)T \)
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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

−17.4059273648664819755120051462, −17.027989869481019779490614292439, −16.16552092811752798783899725300, −15.66551943940804275356388124378, −14.92721570072737254869151671899, −14.41531491024797034705881177507, −13.61760699970058080921576881980, −12.95896163832227064189607131571, −12.56384597405384713251869740293, −12.07116865196566670834003085094, −11.116901022489597363418551273518, −10.01746907126390814299487583352, −9.47165951072387418434863140869, −8.43101490181987764788055263874, −8.107870377491071979666105515481, −7.541685123243836177074203053570, −6.829489087072700312207582461442, −6.29083803808396365765529493111, −5.57901954390031041849388393703, −4.59550252215118224200080276734, −3.83188860812991190047058077028, −3.34867294900407297716045674158, −2.40038099126421311747989752741, −1.10306582671452548744139806886, −0.2634783685762453661810110938, 0.61625428001120936938323573084, 2.29309541188137789112270960149, 2.740966487220099786117026342742, 3.18711344993597314689567050737, 4.29989074852275150960985649240, 4.45719654446246494495562510622, 5.24073937625258568367449727673, 6.33706817832445061596964510749, 6.96925849268325260528474096963, 8.22103882097654151194969389495, 8.603369989505902890356913341929, 9.54541704710635592043172464095, 9.72199409941866485279400335222, 10.68915824651500941270961066234, 11.15614525959971532048080059170, 11.92349459619783941251457165025, 12.30167290168674652338993039281, 13.21828614498492694926282097481, 13.94723370214954815753780291977, 14.56425047030141617417225036859, 15.255041392468756726918725591533, 15.64364978442719176700569144245, 16.46860994177233563109786754682, 17.04808664037651085675361284419, 18.16692049360118567153803547858

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