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Properties

Label 1-6001-6001.2370-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.580 - 0.813i$
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  − 2-s + (0.831 − 0.555i)3-s + 4-s + (0.980 + 0.195i)5-s + (−0.831 + 0.555i)6-s + (0.195 + 0.980i)7-s − 8-s + (0.382 − 0.923i)9-s + (−0.980 − 0.195i)10-s + (0.923 − 0.382i)11-s + (0.831 − 0.555i)12-s + (−0.980 + 0.195i)13-s + (−0.195 − 0.980i)14-s + (0.923 − 0.382i)15-s + 16-s + ⋯
L(s)  = 1  − 2-s + (0.831 − 0.555i)3-s + 4-s + (0.980 + 0.195i)5-s + (−0.831 + 0.555i)6-s + (0.195 + 0.980i)7-s − 8-s + (0.382 − 0.923i)9-s + (−0.980 − 0.195i)10-s + (0.923 − 0.382i)11-s + (0.831 − 0.555i)12-s + (−0.980 + 0.195i)13-s + (−0.195 − 0.980i)14-s + (0.923 − 0.382i)15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.580 - 0.813i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (2370, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.580 - 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.885271237 - 0.9706774619i\)
\(L(\frac12)\) \(\approx\) \(1.885271237 - 0.9706774619i\)
\(L(1)\) \(\approx\) \(1.167918308 - 0.2134060430i\)
\(L(1)\) \(\approx\) \(1.167918308 - 0.2134060430i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.831 - 0.555i)T \)
5 \( 1 + (0.980 + 0.195i)T \)
7 \( 1 + (0.195 + 0.980i)T \)
11 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 + (-0.980 + 0.195i)T \)
19 \( 1 + (0.923 - 0.382i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (0.923 + 0.382i)T \)
31 \( 1 + (-0.980 - 0.195i)T \)
37 \( 1 + (0.831 + 0.555i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (-0.382 - 0.923i)T \)
47 \( 1 + (-0.923 + 0.382i)T \)
53 \( 1 + (-0.195 + 0.980i)T \)
59 \( 1 + (0.195 - 0.980i)T \)
61 \( 1 + (-0.382 - 0.923i)T \)
67 \( 1 + (-0.195 - 0.980i)T \)
71 \( 1 + (0.980 - 0.195i)T \)
73 \( 1 + (0.382 - 0.923i)T \)
79 \( 1 + (0.980 - 0.195i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.195 - 0.980i)T \)
97 \( 1 + (-0.923 + 0.382i)T \)
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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

−17.858353904299013503027857451758, −17.14647420355940526813662975422, −16.47346858609073215445007599939, −16.25591530372903640262708672235, −15.07596818085245955785930241286, −14.61066294705920390006871543409, −14.07418402118795783168867693030, −13.37656086286885407953859311935, −12.54038944667104737456142023319, −11.68836114267202571372424963079, −10.9574862674203458359113009354, −10.046470194759141406110749675772, −9.786576341726064378869494228501, −9.4773077539238468685266873420, −8.49753858687553738160257636988, −7.9047475429058700772119087062, −7.20884773140835562116513481010, −6.643541043141657078018510787, −5.617947647681068936696158915970, −4.866306772435454483355718239304, −3.97476165338560751512364790127, −3.201226596541314749569570485, −2.37116967728339042063245474586, −1.643551076088666066134415184409, −1.05669815453013682188135639779, 0.65713456921703117883142368263, 1.79882473091342947185410110696, 1.95193716113191325726748218463, 2.887007336755961167635416720478, 3.34583155971966495220836318311, 4.778023697293236532343485750330, 5.679636367848286172621121861410, 6.46848365469859912007284260370, 6.78187115403834812102735097306, 7.7223906621078187712555945112, 8.34041212579200352760154941597, 9.181451952227449057934117375094, 9.31069025914510949454859347863, 10.000715451454616210270776716651, 10.88298653132040985089806010009, 11.85083881669279957847508831073, 12.15964473963207428583144754103, 12.912625560031405919737065329115, 13.972385638056052301883278348884, 14.3288162089563623373808868620, 14.9571102776971805163448298380, 15.61456049765844039289137568333, 16.49273323488131130200755252378, 17.15803199794388852675254484743, 17.77055303459280539245502512074

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