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Properties

Label 1-6001-6001.965-r0-0-0
Degree $1$
Conductor $6001$
Sign $-0.847 + 0.530i$
Analytic cond. $27.8685$
Root an. cond. $27.8685$
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  + (−0.989 + 0.142i)2-s + (0.510 − 0.860i)3-s + (0.959 − 0.281i)4-s + (0.860 + 0.510i)5-s + (−0.382 + 0.923i)6-s + (−0.382 − 0.923i)7-s + (−0.909 + 0.415i)8-s + (−0.479 − 0.877i)9-s + (−0.923 − 0.382i)10-s + (0.654 − 0.755i)11-s + (0.247 − 0.968i)12-s + (0.177 + 0.984i)13-s + (0.510 + 0.860i)14-s + (0.877 − 0.479i)15-s + (0.841 − 0.540i)16-s + ⋯
L(s)  = 1  + (−0.989 + 0.142i)2-s + (0.510 − 0.860i)3-s + (0.959 − 0.281i)4-s + (0.860 + 0.510i)5-s + (−0.382 + 0.923i)6-s + (−0.382 − 0.923i)7-s + (−0.909 + 0.415i)8-s + (−0.479 − 0.877i)9-s + (−0.923 − 0.382i)10-s + (0.654 − 0.755i)11-s + (0.247 − 0.968i)12-s + (0.177 + 0.984i)13-s + (0.510 + 0.860i)14-s + (0.877 − 0.479i)15-s + (0.841 − 0.540i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.847 + 0.530i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (965, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.847 + 0.530i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08352384335 - 0.2908249694i\)
\(L(\frac12)\) \(\approx\) \(-0.08352384335 - 0.2908249694i\)
\(L(1)\) \(\approx\) \(0.7290415363 - 0.2427244017i\)
\(L(1)\) \(\approx\) \(0.7290415363 - 0.2427244017i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (-0.989 + 0.142i)T \)
3 \( 1 + (0.510 - 0.860i)T \)
5 \( 1 + (0.860 + 0.510i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (0.654 - 0.755i)T \)
13 \( 1 + (0.177 + 0.984i)T \)
19 \( 1 + (-0.977 + 0.212i)T \)
23 \( 1 + (-0.0713 + 0.997i)T \)
29 \( 1 + (-0.959 - 0.281i)T \)
31 \( 1 + (-0.447 - 0.894i)T \)
37 \( 1 + (0.681 - 0.731i)T \)
41 \( 1 + (-0.997 + 0.0713i)T \)
43 \( 1 + (-0.0713 - 0.997i)T \)
47 \( 1 + (-0.877 - 0.479i)T \)
53 \( 1 + (-0.247 - 0.968i)T \)
59 \( 1 + (0.923 - 0.382i)T \)
61 \( 1 + (-0.841 + 0.540i)T \)
67 \( 1 + (-0.382 + 0.923i)T \)
71 \( 1 + (-0.778 + 0.627i)T \)
73 \( 1 + (0.959 - 0.281i)T \)
79 \( 1 + (-0.860 + 0.510i)T \)
83 \( 1 + (0.909 + 0.415i)T \)
89 \( 1 + (-0.681 - 0.731i)T \)
97 \( 1 + (0.909 + 0.415i)T \)
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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

−18.05903542812122805417904225892, −17.44034467610052661208249936222, −16.702366929010787567299324681974, −16.37628747394498430693185689381, −15.50756729871146146815691922301, −14.94184502051379626610476126227, −14.57391258325587587561633379030, −13.37204465372353381012311232721, −12.695209242132591906234386008925, −12.25734877875501700410040575399, −11.22674711229421163180127826337, −10.554592116854690499883782681888, −9.94233028154082103093193060424, −9.4613679704538221086092622141, −8.82841546228222339168466311375, −8.48214146659616601924392910688, −7.6616813627422898883518580528, −6.52350284691509674470692100972, −6.08761363166286834381018557247, −5.18954664736182817709036685971, −4.49946292491467154585740553118, −3.38723268811423381089302175375, −2.76212162291764673486208830825, −2.06397490706392338370315286798, −1.40542609723973790433677959763, 0.08810680533085703384715868767, 1.2123123287466172744931761220, 1.80051227765999579652725927363, 2.39416744752250130781877814436, 3.45343279664117591699487629848, 3.88592220406485374365999101870, 5.561121162081624118600880776875, 6.166296847782596218169840935913, 6.72491352444538336342260598665, 7.152033210561727402774440572376, 7.91118906429170109162818569673, 8.7179005855146992684986897126, 9.319216975285139055502979715927, 9.78541741290038581363034862650, 10.62172967978040272487124600719, 11.36936380447335532008760971321, 11.75083603381953625359271162269, 13.022653000961688533180984524327, 13.34394125246571209639827189605, 14.18678216699248012690435906500, 14.55977039310908433458755268409, 15.285311330383417796810398801441, 16.425336742544408155653800570159, 16.850418204708001917581868809, 17.33133192705171836389518018146

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