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Properties

Label 1-6017-6017.1135-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.997 + 0.0684i$
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  + (−0.104 + 0.994i)2-s + (−0.309 + 0.951i)3-s + (−0.978 − 0.207i)4-s + (0.104 + 0.994i)5-s + (−0.913 − 0.406i)6-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s − 10-s + (0.5 − 0.866i)12-s + (0.104 − 0.994i)13-s + (0.669 + 0.743i)14-s + (−0.978 − 0.207i)15-s + (0.913 + 0.406i)16-s + (0.913 + 0.406i)17-s + (0.669 − 0.743i)18-s + ⋯
L(s)  = 1  + (−0.104 + 0.994i)2-s + (−0.309 + 0.951i)3-s + (−0.978 − 0.207i)4-s + (0.104 + 0.994i)5-s + (−0.913 − 0.406i)6-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s − 10-s + (0.5 − 0.866i)12-s + (0.104 − 0.994i)13-s + (0.669 + 0.743i)14-s + (−0.978 − 0.207i)15-s + (0.913 + 0.406i)16-s + (0.913 + 0.406i)17-s + (0.669 − 0.743i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.997 + 0.0684i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.997 + 0.0684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7960883520 + 0.02726898747i\)
\(L(\frac12)\) \(\approx\) \(0.7960883520 + 0.02726898747i\)
\(L(1)\) \(\approx\) \(0.6446426677 + 0.4930692197i\)
\(L(1)\) \(\approx\) \(0.6446426677 + 0.4930692197i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.104 + 0.994i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (0.104 + 0.994i)T \)
7 \( 1 + (0.669 - 0.743i)T \)
13 \( 1 + (0.104 - 0.994i)T \)
17 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + (-0.669 - 0.743i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.669 + 0.743i)T \)
41 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.669 + 0.743i)T \)
53 \( 1 + (-0.104 + 0.994i)T \)
59 \( 1 + (-0.669 + 0.743i)T \)
61 \( 1 + (0.913 + 0.406i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.913 - 0.406i)T \)
73 \( 1 + (-0.669 + 0.743i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.913 + 0.406i)T \)
89 \( 1 - T \)
97 \( 1 + (0.913 - 0.406i)T \)
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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

−17.78020738595589413234622514929, −17.21551846420063255362270732756, −16.71369997571115560356926466383, −15.98427297759013633710812858529, −14.72650822760951005804783501009, −14.259877113450776770067974505591, −13.49111314339040151152989890096, −12.97632430010037032550653646926, −12.25178573962916877516307486657, −11.84037178553332331160666582977, −11.45941364300601449989809126879, −10.5680345188713380804940987326, −9.65434611586324602849884643634, −8.995065957077589497052014738208, −8.42773829603331700193972230524, −7.931796074838581806376148524121, −7.05761194629085505474475817652, −5.94858512156364706681088914698, −5.337481754617554927905542471246, −4.879143604257677011705637882772, −3.95149613802736078976348233996, −3.02871077357941840850113989277, −1.97795342184089910895402286478, −1.69186909671612567217431746067, −0.99145451046904057853445018108, 0.2520845950609981440267177236, 1.29970717969958447448369072739, 2.77336500400244165022589838681, 3.43160318054191223450041066492, 4.23726092548061546353715763775, 4.79630796117542307214575382625, 5.58127450713801041854718196234, 6.20129191824495708819794186369, 6.89306828655449497708291470522, 7.61420368672016129865244338479, 8.32572337374987711052586036656, 8.89378053348621652158810503976, 10.054596849525066397037142302804, 10.273451293768277034174715272916, 10.75888424293554780537937213330, 11.6094506606002354696392562483, 12.503588378323248579118934737046, 13.53831743860553003880377368079, 13.96761220497514757634029795684, 14.73284728448746205633607230090, 15.198768773794095738228365321, 15.48292765476652039096741812965, 16.515251084442201798224877298039, 17.15901996857641402415672593523, 17.395360607481746457240113644057

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