↑

Properties

Label 1-6017-6017.1069-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.962 + 0.271i$
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  + (−0.0356 + 0.999i)2-s + (−0.858 − 0.512i)3-s + (−0.997 − 0.0712i)4-s + (−0.102 + 0.994i)5-s + (0.543 − 0.839i)6-s + (−0.0953 − 0.995i)7-s + (0.106 − 0.994i)8-s + (0.473 + 0.880i)9-s + (−0.990 − 0.137i)10-s + (0.819 + 0.572i)12-s + (−0.712 − 0.701i)13-s + (0.998 − 0.0598i)14-s + (0.597 − 0.801i)15-s + (0.989 + 0.142i)16-s + (−0.835 + 0.549i)17-s + (−0.896 + 0.442i)18-s + ⋯
L(s)  = 1  + (−0.0356 + 0.999i)2-s + (−0.858 − 0.512i)3-s + (−0.997 − 0.0712i)4-s + (−0.102 + 0.994i)5-s + (0.543 − 0.839i)6-s + (−0.0953 − 0.995i)7-s + (0.106 − 0.994i)8-s + (0.473 + 0.880i)9-s + (−0.990 − 0.137i)10-s + (0.819 + 0.572i)12-s + (−0.712 − 0.701i)13-s + (0.998 − 0.0598i)14-s + (0.597 − 0.801i)15-s + (0.989 + 0.142i)16-s + (−0.835 + 0.549i)17-s + (−0.896 + 0.442i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.962 + 0.271i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1069, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.962 + 0.271i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05051448515 + 0.3649328187i\)
\(L(\frac12)\) \(\approx\) \(0.05051448515 + 0.3649328187i\)
\(L(1)\) \(\approx\) \(0.5465598090 + 0.2264140461i\)
\(L(1)\) \(\approx\) \(0.5465598090 + 0.2264140461i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.0356 + 0.999i)T \)
3 \( 1 + (-0.858 - 0.512i)T \)
5 \( 1 + (-0.102 + 0.994i)T \)
7 \( 1 + (-0.0953 - 0.995i)T \)
13 \( 1 + (-0.712 - 0.701i)T \)
17 \( 1 + (-0.835 + 0.549i)T \)
19 \( 1 + (0.786 + 0.617i)T \)
23 \( 1 + (-0.756 - 0.654i)T \)
29 \( 1 + (0.276 + 0.961i)T \)
31 \( 1 + (0.127 - 0.991i)T \)
37 \( 1 + (0.995 + 0.0942i)T \)
41 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (0.895 + 0.444i)T \)
47 \( 1 + (0.247 + 0.968i)T \)
53 \( 1 + (-0.859 - 0.510i)T \)
59 \( 1 + (0.00805 - 0.999i)T \)
61 \( 1 + (-0.995 + 0.0988i)T \)
67 \( 1 + (0.905 - 0.423i)T \)
71 \( 1 + (0.0218 - 0.999i)T \)
73 \( 1 + (0.954 + 0.296i)T \)
79 \( 1 + (0.999 + 0.0414i)T \)
83 \( 1 + (-0.984 - 0.176i)T \)
89 \( 1 + (0.868 + 0.495i)T \)
97 \( 1 + (-0.930 - 0.366i)T \)
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

−17.40272037851213660117496340315, −16.99428235735363286744666880316, −16.00366463754801309985462996361, −15.71332527525541133987070945475, −14.88240994023860345397065116442, −13.85700572920125277530419572775, −13.33849219526886828602691161648, −12.40377464631101647179466695719, −12.03840891817486985611646911809, −11.64208026909249019872833850834, −11.00998759773486088497617859200, −9.99224617797008612925220746831, −9.44256005748788076626395231049, −9.13074546086655483207903328841, −8.37282978531243591268453610333, −7.398870077454469560256777153264, −6.368230216696268759270433874539, −5.48606416538359974488147944561, −5.06827841629260871236398921358, −4.455618904907676138344136165516, −3.820981573220148195000787607865, −2.76305901373054036618238673700, −2.03280882680180101124820053125, −1.126155316086522256265963973248, −0.16587127087205162396450556840, 0.72260881607469808686111344666, 1.77835292790055498268571298686, 2.90760128645031744778003722765, 3.839863611229752185530725265506, 4.51807856353315450835000338274, 5.24357404228543240177076009725, 6.27051418661411697396757213128, 6.373021286017573463629056649148, 7.23421496236844854772762207357, 7.78574864885733868711279581752, 8.1256425701569718972248850297, 9.5856370176901658806452249266, 10.02981788434532718658429634854, 10.73701437263508154746453505611, 11.215040435443770301909518743168, 12.330557010524575563133946979190, 12.77367844784198432490191896921, 13.61190183887228558671620109153, 14.089451146464615901888138516485, 14.73850757515347136328391985670, 15.48567256805925102489926636156, 16.14003908347585887548375985981, 16.784675824417753495221859254434, 17.3358589398728244169331778372, 17.95395802480716258531580590755

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