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Properties

Label 1-6001-6001.5997-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.460 - 0.887i$
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  + (−0.415 + 0.909i)2-s + (−0.415 − 0.909i)3-s + (−0.654 − 0.755i)4-s + (−0.415 − 0.909i)5-s + 6-s − 7-s + (0.959 − 0.281i)8-s + (−0.654 + 0.755i)9-s + 10-s + (−0.540 − 0.841i)11-s + (−0.415 + 0.909i)12-s + (−0.989 − 0.142i)13-s + (0.415 − 0.909i)14-s + (−0.654 + 0.755i)15-s + (−0.142 + 0.989i)16-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)2-s + (−0.415 − 0.909i)3-s + (−0.654 − 0.755i)4-s + (−0.415 − 0.909i)5-s + 6-s − 7-s + (0.959 − 0.281i)8-s + (−0.654 + 0.755i)9-s + 10-s + (−0.540 − 0.841i)11-s + (−0.415 + 0.909i)12-s + (−0.989 − 0.142i)13-s + (0.415 − 0.909i)14-s + (−0.654 + 0.755i)15-s + (−0.142 + 0.989i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.460 - 0.887i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (5997, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.460 - 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2585867548 - 0.1571270133i\)
\(L(\frac12)\) \(\approx\) \(0.2585867548 - 0.1571270133i\)
\(L(1)\) \(\approx\) \(0.4363616936 - 0.05542701702i\)
\(L(1)\) \(\approx\) \(0.4363616936 - 0.05542701702i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (-0.415 + 0.909i)T \)
3 \( 1 + (-0.415 - 0.909i)T \)
5 \( 1 + (-0.415 - 0.909i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.540 - 0.841i)T \)
13 \( 1 + (-0.989 - 0.142i)T \)
19 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (-0.540 + 0.841i)T \)
29 \( 1 + (-0.755 - 0.654i)T \)
31 \( 1 + (-0.841 + 0.540i)T \)
37 \( 1 + (-0.959 + 0.281i)T \)
41 \( 1 + (0.540 + 0.841i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (0.654 + 0.755i)T \)
53 \( 1 + (-0.909 + 0.415i)T \)
59 \( 1 + iT \)
61 \( 1 + (0.989 + 0.142i)T \)
67 \( 1 + iT \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.755 - 0.654i)T \)
79 \( 1 + (-0.415 + 0.909i)T \)
83 \( 1 + (0.959 + 0.281i)T \)
89 \( 1 + (0.281 - 0.959i)T \)
97 \( 1 + (0.281 - 0.959i)T \)
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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

−17.844488531234983714007255536059, −17.28473563913641179046629610228, −16.49716460101594216726313283494, −16.04658900979595257132334010894, −15.26804869885448753976579609505, −14.51400347968256656649343300846, −14.089304832350646668696655904543, −12.74157613838117808689544996881, −12.50289428662620458039851114296, −11.83808989975085238465587759963, −11.01126988315826001376967084292, −10.44659963701060445835079721497, −10.02762405594330210708439927549, −9.51372829290494925787470236626, −8.7862924822771248182063185182, −7.74106597850427633199702106934, −7.22255778525672654193215829646, −6.37592427197543963816146635700, −5.453051916214801490138067306842, −4.664768386935428531983414339921, −3.747710189655306332295310605934, −3.58581146025386488332588051331, −2.51506754217528314989062901262, −2.09303091981412033170248889669, −0.36392693849231298465033697162, 0.28248496050018910407020734500, 1.01476570420127509105617097582, 2.062286259886883274345156439709, 3.044187351420093430632942956870, 4.092688199167152724029016608387, 4.99321547927898182590885192737, 5.592096728770004660842056538427, 6.04748541333309373660198524790, 6.991051916007880482686921483628, 7.50222305573936816020853139589, 8.007760070936437250988652611592, 8.91037569923007127539833551160, 9.28267342863166495891559148640, 10.20173814367645991286989914994, 10.98191023526041205810887896490, 11.7547431955542503139965644121, 12.58584555898132686484201651933, 13.04617964591745598266167329158, 13.5675616500190173052663039211, 14.225989560903027476794783269051, 15.29154273702696054931884316578, 15.83584714469837546741244867555, 16.34509635288179359045449580574, 16.98157108246005607592140836069, 17.424875493716726941644884431804

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