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Properties

Label 1-6027-6027.332-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.317 - 0.948i$
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  + (0.925 + 0.379i)2-s + (0.712 + 0.701i)4-s + (0.193 + 0.981i)5-s + (0.393 + 0.919i)8-s + (−0.193 + 0.981i)10-s + (−0.999 + 0.0299i)11-s + (0.134 + 0.990i)13-s + (0.0149 + 0.999i)16-s + (−0.251 − 0.967i)17-s + (−0.978 − 0.207i)19-s + (−0.550 + 0.834i)20-s + (−0.936 − 0.351i)22-s + (−0.998 + 0.0598i)23-s + (−0.925 + 0.379i)25-s + (−0.251 + 0.967i)26-s + ⋯
L(s)  = 1  + (0.925 + 0.379i)2-s + (0.712 + 0.701i)4-s + (0.193 + 0.981i)5-s + (0.393 + 0.919i)8-s + (−0.193 + 0.981i)10-s + (−0.999 + 0.0299i)11-s + (0.134 + 0.990i)13-s + (0.0149 + 0.999i)16-s + (−0.251 − 0.967i)17-s + (−0.978 − 0.207i)19-s + (−0.550 + 0.834i)20-s + (−0.936 − 0.351i)22-s + (−0.998 + 0.0598i)23-s + (−0.925 + 0.379i)25-s + (−0.251 + 0.967i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.317 - 0.948i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (332, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.317 - 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3146275008 + 0.4372288676i\)
\(L(\frac12)\) \(\approx\) \(-0.3146275008 + 0.4372288676i\)
\(L(1)\) \(\approx\) \(1.202170617 + 0.7179379641i\)
\(L(1)\) \(\approx\) \(1.202170617 + 0.7179379641i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.925 + 0.379i)T \)
5 \( 1 + (0.193 + 0.981i)T \)
11 \( 1 + (-0.999 + 0.0299i)T \)
13 \( 1 + (0.134 + 0.990i)T \)
17 \( 1 + (-0.251 - 0.967i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (-0.998 + 0.0598i)T \)
29 \( 1 + (-0.0448 - 0.998i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (0.887 + 0.460i)T \)
43 \( 1 + (-0.995 - 0.0896i)T \)
47 \( 1 + (0.925 + 0.379i)T \)
53 \( 1 + (0.712 + 0.701i)T \)
59 \( 1 + (0.420 + 0.907i)T \)
61 \( 1 + (0.447 + 0.894i)T \)
67 \( 1 + (0.104 - 0.994i)T \)
71 \( 1 + (-0.0448 + 0.998i)T \)
73 \( 1 + (-0.0747 - 0.997i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (-0.791 - 0.611i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−17.15365680886407323877709150031, −16.35915099702411832483811332544, −15.93159731125919844375920810013, −15.17659338803282252924206655606, −14.66487569159070269787062524120, −13.79404232522092466672407385100, −13.07644271309785745267116137265, −12.73814684958573325196042918135, −12.35064176548472039347138066928, −11.33555707784923195812905315405, −10.61439327659426168426599028204, −10.2269083292095186718253628356, −9.40406235723177357011180415908, −8.33932202618788209965334576920, −8.0690334488419286699333549003, −6.97206830264367422230479494258, −6.15577319914489284833431949714, −5.39916892628053634796359246441, −5.19628266984826630397666695663, −4.093575972643477958691613329809, −3.72464342772009978121195012327, −2.60452084680828569348424526411, −1.998526998231042876922147708366, −1.19431133616464511751575943349, −0.082247281089419295002492343378, 1.79495350757003456844103135152, 2.456176510562811009762622928662, 2.886843881861391227264717823466, 4.02575097659561647073279404902, 4.34817495965282610770061937234, 5.40442175421023556154881221300, 5.992058692644901342328060957791, 6.64545610558824954864488089538, 7.28644408280044587508337252966, 7.8318758203145268731745609223, 8.68564610844075238265934263976, 9.631579084335507728815250381376, 10.38997090165154179750378886215, 11.07199753991614964594864376599, 11.60644473958625082202554296794, 12.233058947069677418042180295269, 13.316738019234037803322479643658, 13.50105591254801251247770135771, 14.21216691473220487460517319449, 14.935440419510658861221742710910, 15.35230493291603001932380530625, 16.103471088218863403523186373606, 16.65528185298049816289769279223, 17.44585931455533781139770226129, 18.21361773038407652437016783928

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