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Properties

Label 1-6027-6027.3002-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.0808 + 0.996i$
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  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)5-s + (−0.222 + 0.974i)8-s + (0.222 + 0.974i)10-s + (0.433 + 0.900i)11-s + (0.433 + 0.900i)13-s + (−0.222 − 0.974i)16-s + (0.781 − 0.623i)17-s + i·19-s + (−0.623 − 0.781i)20-s + (−0.781 − 0.623i)22-s + (−0.623 + 0.781i)23-s + (−0.900 − 0.433i)25-s + (−0.781 − 0.623i)26-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)5-s + (−0.222 + 0.974i)8-s + (0.222 + 0.974i)10-s + (0.433 + 0.900i)11-s + (0.433 + 0.900i)13-s + (−0.222 − 0.974i)16-s + (0.781 − 0.623i)17-s + i·19-s + (−0.623 − 0.781i)20-s + (−0.781 − 0.623i)22-s + (−0.623 + 0.781i)23-s + (−0.900 − 0.433i)25-s + (−0.781 − 0.623i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0808 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0808 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.0808 + 0.996i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (3002, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.0808 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7057942939 + 0.7653446007i\)
\(L(\frac12)\) \(\approx\) \(0.7057942939 + 0.7653446007i\)
\(L(1)\) \(\approx\) \(0.7560252613 + 0.1537286798i\)
\(L(1)\) \(\approx\) \(0.7560252613 + 0.1537286798i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.900 + 0.433i)T \)
5 \( 1 + (0.222 - 0.974i)T \)
11 \( 1 + (0.433 + 0.900i)T \)
13 \( 1 + (0.433 + 0.900i)T \)
17 \( 1 + (0.781 - 0.623i)T \)
19 \( 1 + iT \)
23 \( 1 + (-0.623 + 0.781i)T \)
29 \( 1 + (0.781 - 0.623i)T \)
31 \( 1 - T \)
37 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (0.433 + 0.900i)T \)
53 \( 1 + (-0.781 - 0.623i)T \)
59 \( 1 + (-0.222 - 0.974i)T \)
61 \( 1 + (0.623 + 0.781i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.781 - 0.623i)T \)
73 \( 1 + (-0.900 - 0.433i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (0.433 - 0.900i)T \)
97 \( 1 + iT \)
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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

−17.6756094235985299415169068684, −17.06992559706980592529663096969, −16.297431980935597425074846123647, −15.752873733005891675298791075587, −14.95901889072204635340020043947, −14.32330340901631488648130451845, −13.56173095512381915520221505444, −12.805130850256782817111020437942, −12.14313147581163345147746080174, −11.300968681137604698570295764938, −10.82922558097670873682105612772, −10.34523892950448289386442739818, −9.70023920305053509802357028943, −8.72111099368762367943867492273, −8.42630289025218073805215191274, −7.43037005682004984161718747781, −6.99156683434458666347512168990, −6.032766397455941843197127178931, −5.707547411374563683999486835675, −4.23500621238042408985867419036, −3.45807721374941100018370932135, −2.979307906525753513422501442180, −2.22616528347456953341050395247, −1.28484336571964406589193814172, −0.39731443675778806598364562258, 1.02628250246514557590115893811, 1.573444704711683334195198728886, 2.21443355656465220015834505594, 3.45634700044005036576177510230, 4.43005253506562551625290626195, 4.99740587209157821343351406020, 5.973103137399866630643941760073, 6.29437881655721810489487129626, 7.39254055049355025351038166752, 7.81460829781821082608493751577, 8.56952876771093553992982723711, 9.34889525150421319025644236112, 9.67224857471750263186157551475, 10.25194383822402394436242549621, 11.385869782890792591777718119928, 11.833382972062367957928574472372, 12.45580868275363236011206701580, 13.3544317720372241737520905502, 14.25349016975633698488630094805, 14.49580418148972817710699511898, 15.591983798733523554789047613984, 16.07533812119454834017015359592, 16.61798402249545115563762586461, 17.16272506013810736961301912960, 17.80653938666946089294805319938

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