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Properties

Label 1-6017-6017.3722-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.923 - 0.384i$
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  + (−0.527 − 0.849i)2-s + (−0.809 − 0.587i)3-s + (−0.443 + 0.896i)4-s + (−0.861 + 0.506i)5-s + (−0.0724 + 0.997i)6-s + (0.926 − 0.377i)7-s + (0.995 − 0.0965i)8-s + (0.309 + 0.951i)9-s + (0.885 + 0.464i)10-s + (0.885 − 0.464i)12-s + (0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.995 + 0.0965i)15-s + (−0.607 − 0.794i)16-s + (−0.998 + 0.0483i)17-s + (0.644 − 0.764i)18-s + ⋯
L(s)  = 1  + (−0.527 − 0.849i)2-s + (−0.809 − 0.587i)3-s + (−0.443 + 0.896i)4-s + (−0.861 + 0.506i)5-s + (−0.0724 + 0.997i)6-s + (0.926 − 0.377i)7-s + (0.995 − 0.0965i)8-s + (0.309 + 0.951i)9-s + (0.885 + 0.464i)10-s + (0.885 − 0.464i)12-s + (0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.995 + 0.0965i)15-s + (−0.607 − 0.794i)16-s + (−0.998 + 0.0483i)17-s + (0.644 − 0.764i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.923 - 0.384i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (3722, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.923 - 0.384i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8145546546 - 0.1627583421i\)
\(L(\frac12)\) \(\approx\) \(0.8145546546 - 0.1627583421i\)
\(L(1)\) \(\approx\) \(0.5768480329 - 0.2042661865i\)
\(L(1)\) \(\approx\) \(0.5768480329 - 0.2042661865i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.527 - 0.849i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-0.861 + 0.506i)T \)
7 \( 1 + (0.926 - 0.377i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (-0.998 + 0.0483i)T \)
19 \( 1 + (0.0241 - 0.999i)T \)
23 \( 1 + (0.120 + 0.992i)T \)
29 \( 1 + (0.485 + 0.873i)T \)
31 \( 1 + (0.958 + 0.285i)T \)
37 \( 1 + (0.485 + 0.873i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.885 - 0.464i)T \)
47 \( 1 + (-0.681 + 0.732i)T \)
53 \( 1 + (0.399 - 0.916i)T \)
59 \( 1 + (0.0241 + 0.999i)T \)
61 \( 1 + (-0.0724 + 0.997i)T \)
67 \( 1 + (0.885 - 0.464i)T \)
71 \( 1 + (0.958 - 0.285i)T \)
73 \( 1 + (0.215 - 0.976i)T \)
79 \( 1 + (0.715 + 0.698i)T \)
83 \( 1 + (-0.861 + 0.506i)T \)
89 \( 1 + (-0.970 - 0.239i)T \)
97 \( 1 + (0.779 - 0.626i)T \)
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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

−17.468120421521661782669412470530, −17.1761564852552258086671878714, −16.40339890589046612425553764168, −15.72729420006276447995811960953, −15.43958876483863214627849400432, −14.81585495543780555107837581176, −14.14392970418727180104174260883, −13.056909561257862758633367407383, −12.44123712945952238647071682338, −11.59486983806908797107210520902, −11.097386236333799568981232636262, −10.47002906678757943509905328800, −9.74376152779452222120412266598, −8.93542018970793166903851534415, −8.18883165249183376301834411761, −8.01356109287833252149471551964, −6.94346449130740609722139894660, −6.18717161085981005406683655940, −5.58409091680670613753586140447, −4.804928447807967895564903962251, −4.45509055951877862147128528764, −3.67139358735537870440112016003, −2.308948678437417986115942004159, −1.133913608830686031368919685378, −0.51475695500309621739246715881, 0.67373303553763934975859334552, 1.43141217561712251655473883685, 2.17767519360660685660672140119, 3.010068466519217897235792552221, 4.05049834907988848728284194765, 4.54959032825380421996307009167, 5.167679448050707126587508187485, 6.57720452003416404900424091009, 6.97142194997328394947658406085, 7.59702910539504380505041480503, 8.35447902337873120333218093124, 8.85807949349148392876182700807, 9.93719907084519132361660657355, 10.756127024047986090902654067857, 11.13948044166848548648801303012, 11.589513371261556923205103348, 12.04392944289429786265658842454, 12.901059548476944302972398222132, 13.72409949703958182325361350098, 13.9932863269780260165568228169, 15.22630730692439381001891756769, 15.8308428608088685038565058699, 16.628964814133525238081511791365, 17.2211697439157183337222862208, 17.93693980792911429936584261308

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