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Properties

Label 1-6027-6027.170-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.752 - 0.659i$
Analytic cond. $27.9892$
Root an. cond. $27.9892$
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  + (−0.460 − 0.887i)2-s + (−0.575 + 0.817i)4-s + (−0.237 + 0.971i)5-s + (0.990 + 0.134i)8-s + (0.971 − 0.237i)10-s + (0.0970 − 0.995i)11-s + (−0.674 − 0.738i)13-s + (−0.337 − 0.941i)16-s + (0.938 − 0.344i)17-s + (0.777 − 0.629i)19-s + (−0.657 − 0.753i)20-s + (−0.928 + 0.372i)22-s + (0.193 − 0.981i)23-s + (−0.887 − 0.460i)25-s + (−0.344 + 0.938i)26-s + ⋯
L(s)  = 1  + (−0.460 − 0.887i)2-s + (−0.575 + 0.817i)4-s + (−0.237 + 0.971i)5-s + (0.990 + 0.134i)8-s + (0.971 − 0.237i)10-s + (0.0970 − 0.995i)11-s + (−0.674 − 0.738i)13-s + (−0.337 − 0.941i)16-s + (0.938 − 0.344i)17-s + (0.777 − 0.629i)19-s + (−0.657 − 0.753i)20-s + (−0.928 + 0.372i)22-s + (0.193 − 0.981i)23-s + (−0.887 − 0.460i)25-s + (−0.344 + 0.938i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.752 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.752 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.752 - 0.659i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.752 - 0.659i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3897914064 - 1.036121565i\)
\(L(\frac12)\) \(\approx\) \(0.3897914064 - 1.036121565i\)
\(L(1)\) \(\approx\) \(0.7240661493 - 0.3582659715i\)
\(L(1)\) \(\approx\) \(0.7240661493 - 0.3582659715i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.460 - 0.887i)T \)
5 \( 1 + (-0.237 + 0.971i)T \)
11 \( 1 + (0.0970 - 0.995i)T \)
13 \( 1 + (-0.674 - 0.738i)T \)
17 \( 1 + (0.938 - 0.344i)T \)
19 \( 1 + (0.777 - 0.629i)T \)
23 \( 1 + (0.193 - 0.981i)T \)
29 \( 1 + (0.969 - 0.244i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (0.0149 - 0.999i)T \)
43 \( 1 + (0.880 + 0.473i)T \)
47 \( 1 + (0.953 + 0.301i)T \)
53 \( 1 + (0.171 + 0.985i)T \)
59 \( 1 + (-0.525 - 0.850i)T \)
61 \( 1 + (0.323 - 0.946i)T \)
67 \( 1 + (0.0523 - 0.998i)T \)
71 \( 1 + (0.244 - 0.969i)T \)
73 \( 1 + (-0.149 - 0.988i)T \)
79 \( 1 + (-0.965 - 0.258i)T \)
83 \( 1 + (-0.623 - 0.781i)T \)
89 \( 1 + (-0.215 + 0.976i)T \)
97 \( 1 + (0.891 - 0.453i)T \)
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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

−17.548828438303116856891194257177, −17.3522893010492621671993274898, −16.66769265203845358009808766777, −16.056237001353889041681393657510, −15.49576024566987334806061589048, −14.815868249789060605100972104274, −14.13943626515561595644799181503, −13.55475740759784848452187416636, −12.70086026961525878836908211449, −12.037899794808933480232773449622, −11.56847429080745834079888259604, −10.1687621532291693393725955407, −9.939935905593111850680776665079, −9.25523481830412919116463497969, −8.546750745971887871226054127662, −7.87662438840690118211429514494, −7.30502185566572789573757244430, −6.69220930389128381390939189191, −5.58669644254881113567835988021, −5.28973895824576355168250119489, −4.37356798289781432301703553302, −3.9684407394187972928015983988, −2.61147110420979208745949443790, −1.43968835104572149480581233378, −1.11523241477355393021793122216, 0.41708618763058672644405995864, 1.05892038579992102331875058339, 2.39770292976830825187032675937, 2.90671705579308777962356546382, 3.31061341466399866235492946419, 4.27895479051708843489038879905, 5.06092154410033963841609635466, 5.96797902032275667848237793681, 6.81028106907221455364668610990, 7.70169907117996193658174100168, 7.94494592261887209533504505501, 8.9357842081317668235763912316, 9.58849644655333180922275423918, 10.36108439734080503320382759326, 10.741870574216487390866449829242, 11.41415923528307965621487174723, 12.13222510662681524854806417801, 12.54535589417490259433273765016, 13.64550501944336628032097834582, 14.03196045460376549843043637976, 14.63809499797047172892272470238, 15.68352193436329463668504804400, 16.12039100238577665965224256077, 17.08816862216049605149439646132, 17.540453449051315635135860138100

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