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Properties

Label 1-6001-6001.6-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.223 - 0.974i$
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 + (−0.382 − 0.923i)11-s + (−0.831 + 0.555i)12-s + (−0.195 − 0.980i)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 + (−0.382 − 0.923i)11-s + (−0.831 + 0.555i)12-s + (−0.195 − 0.980i)13-s + (−0.195 − 0.980i)14-s + (0.923 − 0.382i)15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.223 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.223 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.223 - 0.974i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.223 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.872431908 - 1.491877122i\)
\(L(\frac12)\) \(\approx\) \(1.872431908 - 1.491877122i\)
\(L(1)\) \(\approx\) \(1.376552672 - 0.2902203657i\)
\(L(1)\) \(\approx\) \(1.376552672 - 0.2902203657i\)

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 \( 1 + (-0.382 - 0.923i)T \)
13 \( 1 + (-0.195 - 0.980i)T \)
19 \( 1 + (0.923 - 0.382i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (-0.382 + 0.923i)T \)
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 \( 1 + (-0.980 - 0.195i)T \)
59 \( 1 + (0.980 + 0.195i)T \)
61 \( 1 + (-0.923 + 0.382i)T \)
67 \( 1 + (0.980 - 0.195i)T \)
71 \( 1 + (0.980 - 0.195i)T \)
73 \( 1 + (0.923 + 0.382i)T \)
79 \( 1 + (-0.980 + 0.195i)T \)
83 \( 1 + iT \)
89 \( 1 + (0.980 - 0.195i)T \)
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

−17.87364524692763387811966509626, −17.067138466123876722719970329448, −16.33361677208628137457921274470, −15.772445805981223098052853145288, −15.346914548596621280583469104001, −14.61278021599028672782397552684, −13.89396978253023388503900147357, −12.96137608639252961556418469210, −12.59804703916440862546509929360, −11.90820595731366764550060037675, −11.50401840715693344206948897158, −11.09868744021645158073456107446, −10.01039406456499554693421331157, −9.3677882369090796220530407326, −8.007207586863082573249944295115, −7.6437897577512428497860528747, −6.95937367818676723042251241773, −6.2887916305564559739511317984, −5.67279831062695041687226253183, −4.774443257780655678968201430689, −4.49645476288254856851404394336, −3.45022027189951195843874275640, −2.5866839938385922940842967232, −1.99498637742188293203755948779, −0.993117503336574440985197667422, 0.657335643786757717649697723656, 0.98995761581960081418327692655, 2.73514565624306446306789072260, 3.39638967756322785873014463578, 3.81760602201344782220268837137, 4.755731807277597429832505802041, 5.08799580496350698896277291414, 5.87803074015537533738408929072, 6.72681517585139957058303981497, 7.28607149441453170830960719478, 7.95087055490154738502756432038, 8.844869454027137739371307784900, 10.01057611399573140242934267559, 10.54155630188768254017335445960, 11.14382968485486726384530745295, 11.48950425283317073931808115779, 12.484851173484043784822550205354, 12.75683781413481905961157111505, 13.6178533957665561948791425762, 14.29284546139713763079892831425, 15.16109551186696463805667176581, 15.6387622813810383614053081810, 16.11712572575719132484891417504, 16.78445148235695942729213667846, 17.14017297205442280044422525735

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