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Properties

Label 1-6027-6027.2852-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.643 - 0.765i$
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  + (0.946 − 0.323i)2-s + (0.791 − 0.611i)4-s + (−0.163 + 0.986i)5-s + (0.550 − 0.834i)8-s + (0.163 + 0.986i)10-s + (−0.873 − 0.486i)11-s + (0.753 + 0.657i)13-s + (0.251 − 0.967i)16-s + (0.925 − 0.379i)17-s + (0.913 − 0.406i)19-s + (0.473 + 0.880i)20-s + (−0.983 − 0.178i)22-s + (−0.525 − 0.850i)23-s + (−0.946 − 0.323i)25-s + (0.925 + 0.379i)26-s + ⋯
L(s)  = 1  + (0.946 − 0.323i)2-s + (0.791 − 0.611i)4-s + (−0.163 + 0.986i)5-s + (0.550 − 0.834i)8-s + (0.163 + 0.986i)10-s + (−0.873 − 0.486i)11-s + (0.753 + 0.657i)13-s + (0.251 − 0.967i)16-s + (0.925 − 0.379i)17-s + (0.913 − 0.406i)19-s + (0.473 + 0.880i)20-s + (−0.983 − 0.178i)22-s + (−0.525 − 0.850i)23-s + (−0.946 − 0.323i)25-s + (0.925 + 0.379i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.643 - 0.765i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (2852, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.643 - 0.765i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.073597825 - 1.431867316i\)
\(L(\frac12)\) \(\approx\) \(3.073597825 - 1.431867316i\)
\(L(1)\) \(\approx\) \(1.845260923 - 0.3408248216i\)
\(L(1)\) \(\approx\) \(1.845260923 - 0.3408248216i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.946 - 0.323i)T \)
5 \( 1 + (-0.163 + 0.986i)T \)
11 \( 1 + (-0.873 - 0.486i)T \)
13 \( 1 + (0.753 + 0.657i)T \)
17 \( 1 + (0.925 - 0.379i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (-0.525 - 0.850i)T \)
29 \( 1 + (-0.691 + 0.722i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.280 - 0.959i)T \)
43 \( 1 + (-0.0448 + 0.998i)T \)
47 \( 1 + (0.946 - 0.323i)T \)
53 \( 1 + (0.791 - 0.611i)T \)
59 \( 1 + (0.887 - 0.460i)T \)
61 \( 1 + (0.999 + 0.0299i)T \)
67 \( 1 + (0.978 - 0.207i)T \)
71 \( 1 + (-0.691 - 0.722i)T \)
73 \( 1 + (-0.955 + 0.294i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (-0.193 - 0.981i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−17.55419922604136477346528816975, −16.99340526199013589397957176343, −16.32979957918201298859390143812, −15.66943159503631660775412405248, −15.38951228172144513566253634153, −14.53290858773681545537095940960, −13.67732977796416771108901935481, −13.198285410561763798467692802362, −12.7484058693092966871104549901, −11.899948225806519787772936125668, −11.61674862239256352810705495289, −10.51963405365205779108322416921, −9.96169771378912604019968969699, −9.01924714551330533159563736639, −8.06120700044003764949749155492, −7.8223814861641532863854063569, −7.130972252367131003698591542127, −5.862783456922680055768067560190, −5.57458805998066628978825681128, −5.07826537478703157660563516976, −3.92064626901762453902602720646, −3.74716139001101306005139115391, −2.67366250762625251729960798252, −1.80369514816410401007375298324, −0.96914338594472773909358638918, 0.65452336908877581329189218942, 1.76974778435648530100067550653, 2.50928215468137799417084356161, 3.295082102737320099776514310302, 3.63976487636965479805721597215, 4.600333076792773485636782750601, 5.53057341365638164569556934267, 5.849499498048534703201833479334, 6.90093284784053574168822544243, 7.2157902170049427067290217430, 8.07362737212349601591483808316, 9.03999381667097843673680991543, 9.99864293431512776184052479093, 10.46359802251457865750956474141, 11.23980558334009410224646748230, 11.499733548911193771916075460303, 12.41923981170936442140816350253, 13.041921864723673800173694214084, 13.894003687559835188221773162479, 14.18773084770650691204305489437, 14.79196161860189375792956055436, 15.63709596414966204920566269948, 16.18973214522441480253760469744, 16.52391056936292182230209516994, 18.05670686507140362169298133178

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