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Properties

Label 1-6001-6001.490-r0-0-0
Degree $1$
Conductor $6001$
Sign $-0.915 + 0.403i$
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 + (−0.980 + 0.195i)3-s + 4-s + (0.555 − 0.831i)5-s + (0.980 − 0.195i)6-s + (0.831 − 0.555i)7-s − 8-s + (0.923 − 0.382i)9-s + (−0.555 + 0.831i)10-s + (−0.382 + 0.923i)11-s + (−0.980 + 0.195i)12-s + (−0.555 − 0.831i)13-s + (−0.831 + 0.555i)14-s + (−0.382 + 0.923i)15-s + 16-s + ⋯
L(s)  = 1  − 2-s + (−0.980 + 0.195i)3-s + 4-s + (0.555 − 0.831i)5-s + (0.980 − 0.195i)6-s + (0.831 − 0.555i)7-s − 8-s + (0.923 − 0.382i)9-s + (−0.555 + 0.831i)10-s + (−0.382 + 0.923i)11-s + (−0.980 + 0.195i)12-s + (−0.555 − 0.831i)13-s + (−0.831 + 0.555i)14-s + (−0.382 + 0.923i)15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.915 + 0.403i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (490, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.915 + 0.403i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03820317204 - 0.1814219260i\)
\(L(\frac12)\) \(\approx\) \(-0.03820317204 - 0.1814219260i\)
\(L(1)\) \(\approx\) \(0.5146949164 - 0.1179455119i\)
\(L(1)\) \(\approx\) \(0.5146949164 - 0.1179455119i\)

Euler product

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

−18.14263806715215955501608832132, −17.42451269022755460629383719592, −17.05124588003186893472700144845, −16.362268153527712110215917842, −15.56976362829981881051074827989, −15.01205494417428780910832104974, −14.30493556763430492525708761363, −13.4387101457279140300150465897, −12.657286141920737089290927831123, −11.77274618843870950453360583339, −11.21017875799007064912275720072, −10.985151736898109976030335624657, −10.29472576465481533791595289268, −9.31478785284845635405978162138, −9.02153124652540826563625811874, −7.84717151047830745100860242153, −7.40260778853313030190794790358, −6.62120804769766439943228655361, −6.11282098779953905056111685226, −5.37639723070952307550822108099, −4.76302364166801611046647662626, −3.40346422064285117470419427425, −2.56738548042483515279279714972, −1.86823636382800931572180991125, −1.23303995739717165055434205244, 0.08404798612435753083986284929, 0.95598516551740585925241363850, 1.72971942484954477247482618013, 2.29032817260513209002107233562, 3.680019346344997632173570780926, 4.55784991919880531914837208910, 5.28492079883184992630563811803, 5.65262563605265777846993642140, 6.73068039397296216305238081056, 7.25703718820064207908029672707, 8.01912767687244977680895936550, 8.6295192474740900854662074717, 9.56781832028833346287575330899, 10.12816917867210853458455267569, 10.53027492500140444059657765287, 11.16431664755163923706022419033, 12.20116394087223900523052449820, 12.38292633379087477265082507821, 13.11621754747681216987883903824, 14.206525618261600189706056033805, 15.058300716787976285730834051558, 15.4830068925651566669063463296, 16.43225970876563769527469326055, 16.88705158792785561964203058780, 17.30553067593855026021869328440

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