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Properties

Label 1-6017-6017.3254-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: $\chi_{6017} (3254, \cdot )$
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

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

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