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Properties

Label 1-3024-3024.1291-r0-0-0
Degree $1$
Conductor $3024$
Sign $0.310 - 0.950i$
Analytic cond. $14.0433$
Root an. cond. $14.0433$
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  + (0.984 − 0.173i)5-s + (−0.984 − 0.173i)11-s + (−0.984 + 0.173i)13-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.766 − 0.642i)23-s + (0.939 − 0.342i)25-s + (−0.984 − 0.173i)29-s + (0.173 + 0.984i)31-s + i·37-s + (0.173 + 0.984i)41-s + (0.642 − 0.766i)43-s + (0.173 − 0.984i)47-s + (0.866 − 0.5i)53-s − 55-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)5-s + (−0.984 − 0.173i)11-s + (−0.984 + 0.173i)13-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.766 − 0.642i)23-s + (0.939 − 0.342i)25-s + (−0.984 − 0.173i)29-s + (0.173 + 0.984i)31-s + i·37-s + (0.173 + 0.984i)41-s + (0.642 − 0.766i)43-s + (0.173 − 0.984i)47-s + (0.866 − 0.5i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.310 - 0.950i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ 0.310 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.235523396 - 0.8960864475i\)
\(L(\frac12)\) \(\approx\) \(1.235523396 - 0.8960864475i\)
\(L(1)\) \(\approx\) \(1.097896471 - 0.1602597991i\)
\(L(1)\) \(\approx\) \(1.097896471 - 0.1602597991i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.984 - 0.173i)T \)
11 \( 1 + (-0.984 - 0.173i)T \)
13 \( 1 + (-0.984 + 0.173i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (-0.984 - 0.173i)T \)
31 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (0.342 - 0.939i)T \)
61 \( 1 + (0.984 + 0.173i)T \)
67 \( 1 + (0.642 + 0.766i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.984 - 0.173i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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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

−19.10956235207607520916949645040, −18.52552639023407521699816331939, −17.58865166555995568841420816869, −17.26809000461348797634716533534, −16.59885343582147151190746971683, −15.55151189463409401177484984036, −14.93091800258324367866589393432, −14.39172951612488635604559236471, −13.44859310323232416097225370389, −12.85633094291526237342863146231, −12.44450634067126089152791566285, −11.135785384136625343415144597477, −10.6988820036316597438095362957, −9.87982814234991800211187414759, −9.37064553342563594855629831780, −8.479021130631104647173828898832, −7.5337414392263500371644500141, −7.03149034178357744044746078878, −5.87958345047587765417710297265, −5.52352943383547501672406365774, −4.65105906383700512048801956344, −3.6736713192093039655521695227, −2.48444239446872669868560314997, −2.26397234443711780918408859258, −1.01451067894450083262741048139, 0.48412235367315869947870082701, 1.70992602288522478777817585964, 2.48350333979021017376935799315, 3.11333948172992703842545142395, 4.40157074705746832993986011448, 5.17111389417571685033020327273, 5.60752526289206942118586893321, 6.66558616297522194599316362044, 7.248954267064845193306200304203, 8.25047229303526793597143094407, 8.88190584157286407194486307924, 9.8610972963039265969300135379, 10.1550367569959051083595644693, 11.01499516555741954542281505761, 11.9262450327274229480087620718, 12.76558119490423320593008158198, 13.15966333939110879943595138222, 14.06403697963582643202419335422, 14.59810075259616569140821469261, 15.348035415595136745852731025026, 16.33706908102997615198436249171, 16.83005356513914161689933508940, 17.42892126163641951491846932989, 18.34488102302531723443487395524, 18.70613153235516916722504975235

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