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Properties

Label 1-6017-6017.4-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.977 - 0.212i$
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  + (0.287 + 0.957i)2-s + (0.473 + 0.880i)3-s + (−0.835 + 0.549i)4-s + (−0.961 + 0.274i)5-s + (−0.707 + 0.706i)6-s + (0.170 − 0.985i)7-s + (−0.766 − 0.642i)8-s + (−0.550 + 0.834i)9-s + (−0.539 − 0.842i)10-s + (−0.880 − 0.474i)12-s + (0.0149 + 0.999i)13-s + (0.992 − 0.119i)14-s + (−0.697 − 0.716i)15-s + (0.395 − 0.918i)16-s + (−0.603 + 0.797i)17-s + (−0.957 − 0.288i)18-s + ⋯
L(s)  = 1  + (0.287 + 0.957i)2-s + (0.473 + 0.880i)3-s + (−0.835 + 0.549i)4-s + (−0.961 + 0.274i)5-s + (−0.707 + 0.706i)6-s + (0.170 − 0.985i)7-s + (−0.766 − 0.642i)8-s + (−0.550 + 0.834i)9-s + (−0.539 − 0.842i)10-s + (−0.880 − 0.474i)12-s + (0.0149 + 0.999i)13-s + (0.992 − 0.119i)14-s + (−0.697 − 0.716i)15-s + (0.395 − 0.918i)16-s + (−0.603 + 0.797i)17-s + (−0.957 − 0.288i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.977 - 0.212i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.977 - 0.212i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06044177780 + 0.006484356030i\)
\(L(\frac12)\) \(\approx\) \(-0.06044177780 + 0.006484356030i\)
\(L(1)\) \(\approx\) \(0.5164612610 + 0.6833581102i\)
\(L(1)\) \(\approx\) \(0.5164612610 + 0.6833581102i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.287 + 0.957i)T \)
3 \( 1 + (0.473 + 0.880i)T \)
5 \( 1 + (-0.961 + 0.274i)T \)
7 \( 1 + (0.170 - 0.985i)T \)
13 \( 1 + (0.0149 + 0.999i)T \)
17 \( 1 + (-0.603 + 0.797i)T \)
19 \( 1 + (-0.935 + 0.353i)T \)
23 \( 1 + (0.895 + 0.444i)T \)
29 \( 1 + (-0.918 - 0.396i)T \)
31 \( 1 + (-0.367 + 0.930i)T \)
37 \( 1 + (0.403 + 0.914i)T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (-0.980 - 0.194i)T \)
47 \( 1 + (-0.892 - 0.450i)T \)
53 \( 1 + (0.831 + 0.555i)T \)
59 \( 1 + (-0.136 + 0.990i)T \)
61 \( 1 + (-0.531 - 0.847i)T \)
67 \( 1 + (0.0287 + 0.999i)T \)
71 \( 1 + (-0.0770 - 0.997i)T \)
73 \( 1 + (0.465 + 0.884i)T \)
79 \( 1 + (-0.800 + 0.598i)T \)
83 \( 1 + (0.991 - 0.128i)T \)
89 \( 1 + (-0.985 + 0.171i)T \)
97 \( 1 + (-0.547 - 0.837i)T \)
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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

−17.405797856531845219865403656174, −16.40485507029454765240274778752, −15.42563200067955191551951512461, −14.76463719724968264618937386520, −14.74924204879674954496648550554, −13.398186645404199145113016911788, −12.98497586482859225123756690939, −12.55601011595765649678252036592, −11.90778299630253242020424704130, −11.17433586569764998886488585988, −10.91941759860518301582205122882, −9.50285720926937119441154526433, −9.10616675392018692145774626164, −8.40561597688478580700067958482, −7.936868916180430872506657364987, −7.0383285528012713103077262709, −6.15230598144400643177981748522, −5.36651493383446896174354909597, −4.69759427351250858995024627382, −3.827793040278896380818332879581, −3.019845242586315386819582665105, −2.54860453626061944626228063651, −1.78672659267366643917063738563, −0.77265232314847713213894473277, −0.016732995714438732800402907435, 1.5413390848452930756352882260, 2.792354668339387604145246572035, 3.678033937499181789412276270267, 4.049898484549079901884174479087, 4.52406804442300508194277566275, 5.28452715025303747975461417979, 6.36947273173951661320832936560, 6.98071188580068560941242858534, 7.59742007298840430699517296290, 8.34345761191190308402341590797, 8.758517427029296660716020529064, 9.566284641757218210427207132038, 10.39246087689909781875852117865, 11.031221762491801689546078678046, 11.62505891536196109326531260549, 12.66114216452500561996474172824, 13.34797817784772229703840829725, 13.96697385000783155317694725686, 14.67316745711829883020624191817, 15.05562147575476090446249370415, 15.56632015385406249134552982728, 16.524603212742289321031353721540, 16.671156857151935425715638045764, 17.30894818878926315210137040765, 18.28593131203368040098478121454

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