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Properties

Label 1-6027-6027.1073-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.792 - 0.609i$
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  + (0.323 − 0.946i)2-s + (−0.791 − 0.611i)4-s + (−0.986 + 0.163i)5-s + (−0.834 + 0.550i)8-s + (−0.163 + 0.986i)10-s + (0.961 + 0.273i)11-s + (−0.997 − 0.0672i)13-s + (0.251 + 0.967i)16-s + (−0.922 + 0.386i)17-s + (−0.933 + 0.358i)19-s + (0.880 + 0.473i)20-s + (0.569 − 0.822i)22-s + (0.525 − 0.850i)23-s + (0.946 − 0.323i)25-s + (−0.386 + 0.922i)26-s + ⋯
L(s)  = 1  + (0.323 − 0.946i)2-s + (−0.791 − 0.611i)4-s + (−0.986 + 0.163i)5-s + (−0.834 + 0.550i)8-s + (−0.163 + 0.986i)10-s + (0.961 + 0.273i)11-s + (−0.997 − 0.0672i)13-s + (0.251 + 0.967i)16-s + (−0.922 + 0.386i)17-s + (−0.933 + 0.358i)19-s + (0.880 + 0.473i)20-s + (0.569 − 0.822i)22-s + (0.525 − 0.850i)23-s + (0.946 − 0.323i)25-s + (−0.386 + 0.922i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.792 - 0.609i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (1073, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.792 - 0.609i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7402546949 - 0.2517104022i\)
\(L(\frac12)\) \(\approx\) \(0.7402546949 - 0.2517104022i\)
\(L(1)\) \(\approx\) \(0.6994679085 - 0.3573629687i\)
\(L(1)\) \(\approx\) \(0.6994679085 - 0.3573629687i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.323 - 0.946i)T \)
5 \( 1 + (-0.986 + 0.163i)T \)
11 \( 1 + (0.961 + 0.273i)T \)
13 \( 1 + (-0.997 - 0.0672i)T \)
17 \( 1 + (-0.922 + 0.386i)T \)
19 \( 1 + (-0.933 + 0.358i)T \)
23 \( 1 + (0.525 - 0.850i)T \)
29 \( 1 + (0.0224 - 0.999i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.280 + 0.959i)T \)
43 \( 1 + (-0.998 + 0.0448i)T \)
47 \( 1 + (-0.897 + 0.440i)T \)
53 \( 1 + (-0.126 - 0.991i)T \)
59 \( 1 + (-0.887 - 0.460i)T \)
61 \( 1 + (0.0299 + 0.999i)T \)
67 \( 1 + (0.838 - 0.544i)T \)
71 \( 1 + (-0.999 + 0.0224i)T \)
73 \( 1 + (0.294 - 0.955i)T \)
79 \( 1 + (0.258 - 0.965i)T \)
83 \( 1 + (0.222 + 0.974i)T \)
89 \( 1 + (0.830 - 0.557i)T \)
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

−17.4875488260641896682693752405, −17.04445561292409952158639025251, −16.39728546765974397918660595934, −15.78933363071217288116834950516, −15.15544235233620508431329358142, −14.63142679897265576685200740193, −14.08639823429342354086637887954, −13.15466115390841613597157127138, −12.631825027833733886043723127121, −11.96441899015817298190287110022, −11.353497754005661106613560535624, −10.5880694860521340090332369309, −9.363430876358591152717347185743, −9.01794514132296938945713118584, −8.38286851274281833563782386202, −7.55474433811545463291040286963, −6.89780017954680492080216560998, −6.63111485828085146287954100448, −5.42837472841646201312309925846, −4.86614919859218643465353969792, −4.2153148029891303233881398667, −3.55566972035187856857605971671, −2.832543923820604961787207519561, −1.58069426520836152774402205978, −0.330334888713416932398828524880, 0.51301750090630412615122956103, 1.702268686255061816083406189555, 2.33295836965345589975185315494, 3.21284377710905467054056964903, 3.93444667808043790308857549136, 4.51391962711760389550955436728, 4.979667546292272492782226815711, 6.29715558447776114536331564209, 6.63637887334653712491857726744, 7.71293771583956353034923299008, 8.41583304687884133815576670356, 9.04981545498382377841402920877, 9.80654305280430351541227740235, 10.490824963398752776616336733379, 11.17092350526377192073774927277, 11.73038154053909808383734881699, 12.27497718446183860740671807117, 12.875218032659892649336787845076, 13.52878317982147682880006666749, 14.5953780406407559499725238028, 14.855215125103711207843799214458, 15.29566232882714831727800997235, 16.43670017171965878863666989368, 17.098997019186469831506567247460, 17.66019671202175095064165808951

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