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Properties

Label 1-6027-6027.1094-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.969 + 0.246i$
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  + (0.981 + 0.193i)2-s + (0.925 + 0.379i)4-s + (−0.635 + 0.772i)5-s + (0.834 + 0.550i)8-s + (−0.772 + 0.635i)10-s + (−0.696 + 0.717i)11-s + (0.0672 + 0.997i)13-s + (0.712 + 0.701i)16-s + (−0.991 + 0.126i)17-s + (0.629 + 0.777i)19-s + (−0.880 + 0.473i)20-s + (−0.822 + 0.569i)22-s + (−0.999 + 0.0299i)23-s + (−0.193 − 0.981i)25-s + (−0.126 + 0.991i)26-s + ⋯
L(s)  = 1  + (0.981 + 0.193i)2-s + (0.925 + 0.379i)4-s + (−0.635 + 0.772i)5-s + (0.834 + 0.550i)8-s + (−0.772 + 0.635i)10-s + (−0.696 + 0.717i)11-s + (0.0672 + 0.997i)13-s + (0.712 + 0.701i)16-s + (−0.991 + 0.126i)17-s + (0.629 + 0.777i)19-s + (−0.880 + 0.473i)20-s + (−0.822 + 0.569i)22-s + (−0.999 + 0.0299i)23-s + (−0.193 − 0.981i)25-s + (−0.126 + 0.991i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.969 + 0.246i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (1094, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.969 + 0.246i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3126190303 + 2.495357432i\)
\(L(\frac12)\) \(\approx\) \(0.3126190303 + 2.495357432i\)
\(L(1)\) \(\approx\) \(1.395632691 + 0.8543514359i\)
\(L(1)\) \(\approx\) \(1.395632691 + 0.8543514359i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.981 + 0.193i)T \)
5 \( 1 + (-0.635 + 0.772i)T \)
11 \( 1 + (-0.696 + 0.717i)T \)
13 \( 1 + (0.0672 + 0.997i)T \)
17 \( 1 + (-0.991 + 0.126i)T \)
19 \( 1 + (0.629 + 0.777i)T \)
23 \( 1 + (-0.999 + 0.0299i)T \)
29 \( 1 + (0.999 - 0.0224i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (0.971 + 0.237i)T \)
43 \( 1 + (0.998 + 0.0448i)T \)
47 \( 1 + (0.557 + 0.830i)T \)
53 \( 1 + (-0.386 - 0.922i)T \)
59 \( 1 + (0.842 + 0.538i)T \)
61 \( 1 + (-0.850 - 0.525i)T \)
67 \( 1 + (0.998 + 0.0523i)T \)
71 \( 1 + (-0.0224 + 0.999i)T \)
73 \( 1 + (-0.680 + 0.733i)T \)
79 \( 1 + (-0.258 + 0.965i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (0.440 + 0.897i)T \)
97 \( 1 + (0.453 + 0.891i)T \)
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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

−17.374780590775588339229514780380, −16.40668256932188651436720729822, −15.823269736510699569277792721966, −15.61062814973380088123252147901, −14.89342603818037601941096286096, −13.80234087355982050300161793398, −13.51587382332042427776512528843, −12.85699662334691010566375848776, −12.19596161382141879811124127609, −11.6224406321855312282300991021, −10.93616680090724275840681484835, −10.38786641585545510714436677544, −9.47480215489155636916114635158, −8.581116908311960395643353870921, −7.91508587078557250991349460382, −7.383184663094209178668819439797, −6.35529214687448934910328759710, −5.74904108839244191672909283914, −5.013084819199382367192602193, −4.482787044441234877026367432925, −3.74630700460954419679718058580, −2.88920862278002590327678326270, −2.392821891532450898282525740497, −1.11967442420378804890777827558, −0.445468400836106317413890708893, 1.371337525046478537087968241640, 2.44623937848397229972033277519, 2.6857227913922110355630658743, 3.89167368350281714223176112192, 4.21107154302603058465347724272, 4.93614708519170044476330731389, 5.91442976744508540021954937237, 6.57336822223466526632817094945, 7.04718857572617334965054284658, 7.89655356245817094368818878231, 8.24778064276356195338722135796, 9.51921782597996202688648615397, 10.27575110437042256685377876406, 10.87589124940324620562417745671, 11.64743148315683981422793652927, 12.02625025286189098798281913473, 12.745870723875374632125950587950, 13.55697154040634317179853532903, 14.19671967027334308358121740414, 14.585597390893536248829475426919, 15.469874307094609893713838942140, 15.876343829544689768312506630319, 16.28955936850322908388581355164, 17.411929890635821349612640041858, 17.89514839296315194817793625115

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