↑

Properties

Label 1-6017-6017.2781-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.887 + 0.460i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.644 + 0.764i)2-s + (0.309 − 0.951i)3-s + (−0.168 + 0.985i)4-s + (0.215 − 0.976i)5-s + (0.926 − 0.377i)6-s + (−0.527 − 0.849i)7-s + (−0.861 + 0.506i)8-s + (−0.809 − 0.587i)9-s + (0.885 − 0.464i)10-s + (0.885 + 0.464i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.861 − 0.506i)15-s + (−0.943 − 0.331i)16-s + (−0.262 − 0.964i)17-s + (−0.0724 − 0.997i)18-s + ⋯
L(s)  = 1  + (0.644 + 0.764i)2-s + (0.309 − 0.951i)3-s + (−0.168 + 0.985i)4-s + (0.215 − 0.976i)5-s + (0.926 − 0.377i)6-s + (−0.527 − 0.849i)7-s + (−0.861 + 0.506i)8-s + (−0.809 − 0.587i)9-s + (0.885 − 0.464i)10-s + (0.885 + 0.464i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.861 − 0.506i)15-s + (−0.943 − 0.331i)16-s + (−0.262 − 0.964i)17-s + (−0.0724 − 0.997i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.887 + 0.460i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (2781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.887 + 0.460i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2386534563 - 0.9783725125i\)
\(L(\frac12)\) \(\approx\) \(-0.2386534563 - 0.9783725125i\)
\(L(1)\) \(\approx\) \(1.093344672 - 0.3963257890i\)
\(L(1)\) \(\approx\) \(1.093344672 - 0.3963257890i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.644 + 0.764i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.215 - 0.976i)T \)
7 \( 1 + (-0.527 - 0.849i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (-0.262 - 0.964i)T \)
19 \( 1 + (-0.607 - 0.794i)T \)
23 \( 1 + (0.120 - 0.992i)T \)
29 \( 1 + (-0.906 + 0.421i)T \)
31 \( 1 + (0.0241 - 0.999i)T \)
37 \( 1 + (-0.906 + 0.421i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (0.885 + 0.464i)T \)
47 \( 1 + (0.981 + 0.192i)T \)
53 \( 1 + (0.995 - 0.0965i)T \)
59 \( 1 + (-0.607 + 0.794i)T \)
61 \( 1 + (0.926 - 0.377i)T \)
67 \( 1 + (0.885 + 0.464i)T \)
71 \( 1 + (0.0241 + 0.999i)T \)
73 \( 1 + (0.399 - 0.916i)T \)
79 \( 1 + (-0.443 - 0.896i)T \)
83 \( 1 + (0.215 - 0.976i)T \)
89 \( 1 + (-0.970 + 0.239i)T \)
97 \( 1 + (0.836 - 0.548i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.31507718659121540012435423744, −17.43928889653110206448219141312, −16.74529437004279078795950010056, −15.69801848382480598835325499161, −15.33633550711113589051320779862, −14.76401744254816492391172760214, −14.24444519442701939430201888916, −13.655048538323043297562599769693, −12.73866967263416494430270137275, −12.158263684182472702237444295110, −11.367154957309592784191529009827, −10.82802716845820627491263997100, −10.17001879933291359452639839602, −9.642405723982239375709098106077, −9.11420615611838234844758940587, −8.30500388582854852644451186989, −7.1701194510808984405813477849, −6.36052798907298414376978226606, −5.67343833030663164861445508031, −5.229530165134068186882250436969, −4.055975010647524268606812481077, −3.747083111166227042830276140428, −2.93676613850496903743920104707, −2.266628016304886914284895627623, −1.79237902069810466435825410171, 0.21392776264872554546016803127, 0.822371380865938145817447267349, 2.235814756242391676433389597361, 2.69261979258361350647113387569, 3.71619654344087368789747265706, 4.42258280320316814474886867493, 5.15703256408118370884654569852, 5.850814494019789247531984230325, 6.60597434900843757656079715008, 7.3138317241319943865270218195, 7.59436580117242340339878694603, 8.59698047043718441424412360527, 9.018139852120627143614811201897, 9.767051955796693362120573764, 10.86060632064125856557814160378, 11.76423130276970707367252393737, 12.44883923050238504781226478869, 12.89942928910324386334190255894, 13.3548686861491247305976346203, 13.928115194501401638093356769304, 14.566779915982509696975765196058, 15.342311938166687631349408333322, 16.05733415733180397801437401381, 16.77699372337262553465087450584, 17.2724764325773354058143948763

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