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Properties

Label 1-6017-6017.998-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.507 - 0.861i$
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  + (−0.827 − 0.561i)2-s + (0.809 − 0.587i)3-s + (0.369 + 0.929i)4-s + (0.993 − 0.112i)5-s + (−0.999 + 0.0322i)6-s + (−0.984 + 0.176i)7-s + (0.215 − 0.976i)8-s + (0.309 − 0.951i)9-s + (−0.885 − 0.464i)10-s + (0.845 + 0.534i)12-s + (−0.669 − 0.743i)13-s + (0.913 + 0.406i)14-s + (0.737 − 0.675i)15-s + (−0.726 + 0.686i)16-s + (0.152 + 0.988i)17-s + (−0.789 + 0.613i)18-s + ⋯
L(s)  = 1  + (−0.827 − 0.561i)2-s + (0.809 − 0.587i)3-s + (0.369 + 0.929i)4-s + (0.993 − 0.112i)5-s + (−0.999 + 0.0322i)6-s + (−0.984 + 0.176i)7-s + (0.215 − 0.976i)8-s + (0.309 − 0.951i)9-s + (−0.885 − 0.464i)10-s + (0.845 + 0.534i)12-s + (−0.669 − 0.743i)13-s + (0.913 + 0.406i)14-s + (0.737 − 0.675i)15-s + (−0.726 + 0.686i)16-s + (0.152 + 0.988i)17-s + (−0.789 + 0.613i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.507 - 0.861i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (998, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.507 - 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7944103829 - 1.389333359i\)
\(L(\frac12)\) \(\approx\) \(0.7944103829 - 1.389333359i\)
\(L(1)\) \(\approx\) \(0.8799865245 - 0.4980176931i\)
\(L(1)\) \(\approx\) \(0.8799865245 - 0.4980176931i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.827 - 0.561i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (0.993 - 0.112i)T \)
7 \( 1 + (-0.984 + 0.176i)T \)
13 \( 1 + (-0.669 - 0.743i)T \)
17 \( 1 + (0.152 + 0.988i)T \)
19 \( 1 + (-0.184 - 0.982i)T \)
23 \( 1 + (-0.799 + 0.600i)T \)
29 \( 1 + (0.681 - 0.732i)T \)
31 \( 1 + (0.607 + 0.794i)T \)
37 \( 1 + (0.293 + 0.955i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (-0.0402 + 0.999i)T \)
47 \( 1 + (0.513 - 0.857i)T \)
53 \( 1 + (-0.00805 - 0.999i)T \)
59 \( 1 + (-0.184 + 0.982i)T \)
61 \( 1 + (0.999 - 0.0322i)T \)
67 \( 1 + (-0.0402 + 0.999i)T \)
71 \( 1 + (0.384 + 0.923i)T \)
73 \( 1 + (0.414 - 0.910i)T \)
79 \( 1 + (-0.168 - 0.985i)T \)
83 \( 1 + (0.594 + 0.804i)T \)
89 \( 1 + (0.970 + 0.239i)T \)
97 \( 1 + (0.541 + 0.840i)T \)
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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

−17.98635238127500686138642434591, −16.95466284754333057354497564258, −16.700495364796644543562389457676, −16.04275436904372110046582410635, −15.51864637053351586993438309929, −14.49973391928169940573015162316, −14.21238433169070300076926432770, −13.73675870665933290048335579723, −12.81287172030112447535479322530, −11.98725295163382088114989657868, −10.88863729887138973980493621045, −10.25050493288032399355092438320, −9.79537298253091366243662658170, −9.365654833581798961782111857091, −8.81492162284960339946864430634, −7.91585489123495239393820980080, −7.26280523967935478753616998567, −6.51893038478102756136672432413, −5.96153187176399689638618296101, −5.07226769197851741315763435039, −4.378956792701437109762872556360, −3.32447079136928484523284719151, −2.42249303969110788264883506306, −2.09212685780646587518020109675, −0.91356118338109031487088470725, 0.53052251151939679823303540226, 1.36672866294239244922929547958, 2.25909731926039294890753951319, 2.67170065142387392597445651797, 3.34220033421089794634802375658, 4.18325259666662715285164025907, 5.39155495202489877995698082623, 6.39124421040637569294685096693, 6.68091138449231432606116080042, 7.59009787843519633363671439542, 8.31785913722940377662042223795, 8.85813546542951655510901339229, 9.55143631521567743976257481053, 10.102822513563778470204079162444, 10.42829502830049377939539215693, 11.74277961708946246498741421495, 12.275394268510242379942139806650, 12.99105939676948283484420871664, 13.27836221160543692919482539329, 13.96799348452988793360718749640, 14.98571361907471108134074861460, 15.52570866525854973130137418974, 16.30595194585560905511551448903, 17.19164059522239289144467198640, 17.655644338792457140120531919699

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