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Properties

Label 1-6017-6017.4669-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.997 - 0.0663i$
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  + (−0.262 − 0.964i)2-s + (0.309 + 0.951i)3-s + (−0.861 + 0.506i)4-s + (−0.443 − 0.896i)5-s + (0.836 − 0.548i)6-s + (−0.998 + 0.0483i)7-s + (0.715 + 0.698i)8-s + (−0.809 + 0.587i)9-s + (−0.748 + 0.663i)10-s + (−0.748 − 0.663i)12-s + (−0.809 + 0.587i)13-s + (0.309 + 0.951i)14-s + (0.715 − 0.698i)15-s + (0.485 − 0.873i)16-s + (0.926 + 0.377i)17-s + (0.779 + 0.626i)18-s + ⋯
L(s)  = 1  + (−0.262 − 0.964i)2-s + (0.309 + 0.951i)3-s + (−0.861 + 0.506i)4-s + (−0.443 − 0.896i)5-s + (0.836 − 0.548i)6-s + (−0.998 + 0.0483i)7-s + (0.715 + 0.698i)8-s + (−0.809 + 0.587i)9-s + (−0.748 + 0.663i)10-s + (−0.748 − 0.663i)12-s + (−0.809 + 0.587i)13-s + (0.309 + 0.951i)14-s + (0.715 − 0.698i)15-s + (0.485 − 0.873i)16-s + (0.926 + 0.377i)17-s + (0.779 + 0.626i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.997 - 0.0663i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (4669, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.997 - 0.0663i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8226589929 + 0.02734142770i\)
\(L(\frac12)\) \(\approx\) \(0.8226589929 + 0.02734142770i\)
\(L(1)\) \(\approx\) \(0.6964480907 - 0.1292005593i\)
\(L(1)\) \(\approx\) \(0.6964480907 - 0.1292005593i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.262 - 0.964i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.443 - 0.896i)T \)
7 \( 1 + (-0.998 + 0.0483i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (0.926 + 0.377i)T \)
19 \( 1 + (0.981 - 0.192i)T \)
23 \( 1 + (0.568 + 0.822i)T \)
29 \( 1 + (-0.607 - 0.794i)T \)
31 \( 1 + (-0.681 - 0.732i)T \)
37 \( 1 + (-0.607 - 0.794i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.748 - 0.663i)T \)
47 \( 1 + (0.958 + 0.285i)T \)
53 \( 1 + (-0.989 + 0.144i)T \)
59 \( 1 + (0.981 + 0.192i)T \)
61 \( 1 + (0.836 - 0.548i)T \)
67 \( 1 + (-0.748 - 0.663i)T \)
71 \( 1 + (-0.681 + 0.732i)T \)
73 \( 1 + (-0.168 - 0.985i)T \)
79 \( 1 + (0.995 + 0.0965i)T \)
83 \( 1 + (-0.443 - 0.896i)T \)
89 \( 1 + (-0.354 - 0.935i)T \)
97 \( 1 + (0.644 - 0.764i)T \)
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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

−17.89129520791813876953801054501, −17.043625462679236219763904153943, −16.39506333435328492687011223144, −15.79722285995105055501250379871, −14.91891910356714506797250361397, −14.57973499059268291971826771027, −13.94517526164873074238982656375, −13.2807925078218498989074904789, −12.48558973282050451726856308604, −12.1148767423530371423170066017, −11.007800254353047301070317178536, −10.19811737674490834138598255738, −9.67776834471244768358419362352, −8.875918303646523539008123182726, −8.14101864584738997488990871674, −7.33923157774146730148553548632, −7.13934447803239656296424543498, −6.55011583842154928169325481971, −5.66341415036809382463398281512, −5.16337643996078469458273831417, −3.80738237139574191384969896534, −3.208687376121268110925425756830, −2.62502557316807233867580414062, −1.3464612908791944493874575460, −0.42508331573727339164709610357, 0.52847226546778922651611170526, 1.64907964866336184020314877995, 2.5370792904253456202234150758, 3.43176627414294297228279111904, 3.72610646018664743520119771586, 4.54567288286735097770462244174, 5.25020363273295594370890460547, 5.81361749385187505708273982476, 7.37582188702688750378058473081, 7.743009402842707076862484175265, 8.73889916586785761747047513617, 9.30352649400186011307416076510, 9.6317333154643295390131168842, 10.19693163189372742145851958106, 11.161882503932447497770877314370, 11.73586038470881412716114315548, 12.28355490131350505241878004331, 13.071501831702435829004741067027, 13.55799472955566668916903838289, 14.3943904703664960220059467696, 15.11860120519952286979593655611, 15.952811102144353430066577898594, 16.434929479011796640777037448301, 17.02261320162662640424935299760, 17.46627911353871938289248450460

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