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Properties

Label 1-4140-4140.2099-r0-0-0
Degree $1$
Conductor $4140$
Sign $0.452 - 0.891i$
Analytic cond. $19.2260$
Root an. cond. $19.2260$
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  + (0.235 + 0.971i)7-s + (0.981 + 0.189i)11-s + (−0.235 + 0.971i)13-s + (−0.142 − 0.989i)17-s + (0.142 − 0.989i)19-s + (0.786 − 0.618i)29-s + (−0.0475 − 0.998i)31-s + (−0.415 − 0.909i)37-s + (−0.580 − 0.814i)41-s + (0.0475 − 0.998i)43-s + (0.5 + 0.866i)47-s + (−0.888 + 0.458i)49-s + (−0.959 − 0.281i)53-s + (0.235 − 0.971i)59-s + (−0.888 − 0.458i)61-s + ⋯
L(s)  = 1  + (0.235 + 0.971i)7-s + (0.981 + 0.189i)11-s + (−0.235 + 0.971i)13-s + (−0.142 − 0.989i)17-s + (0.142 − 0.989i)19-s + (0.786 − 0.618i)29-s + (−0.0475 − 0.998i)31-s + (−0.415 − 0.909i)37-s + (−0.580 − 0.814i)41-s + (0.0475 − 0.998i)43-s + (0.5 + 0.866i)47-s + (−0.888 + 0.458i)49-s + (−0.959 − 0.281i)53-s + (0.235 − 0.971i)59-s + (−0.888 − 0.458i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.452 - 0.891i$
Analytic conductor: \(19.2260\)
Root analytic conductor: \(19.2260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (2099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4140,\ (0:\ ),\ 0.452 - 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.276312637 - 0.7838063899i\)
\(L(\frac12)\) \(\approx\) \(1.276312637 - 0.7838063899i\)
\(L(1)\) \(\approx\) \(1.086526073 - 0.03984789521i\)
\(L(1)\) \(\approx\) \(1.086526073 - 0.03984789521i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + (0.235 + 0.971i)T \)
11 \( 1 + (0.981 + 0.189i)T \)
13 \( 1 + (-0.235 + 0.971i)T \)
17 \( 1 + (-0.142 - 0.989i)T \)
19 \( 1 + (0.142 - 0.989i)T \)
29 \( 1 + (0.786 - 0.618i)T \)
31 \( 1 + (-0.0475 - 0.998i)T \)
37 \( 1 + (-0.415 - 0.909i)T \)
41 \( 1 + (-0.580 - 0.814i)T \)
43 \( 1 + (0.0475 - 0.998i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.959 - 0.281i)T \)
59 \( 1 + (0.235 - 0.971i)T \)
61 \( 1 + (-0.888 - 0.458i)T \)
67 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.142 - 0.989i)T \)
79 \( 1 + (-0.723 - 0.690i)T \)
83 \( 1 + (-0.580 + 0.814i)T \)
89 \( 1 + (-0.841 - 0.540i)T \)
97 \( 1 + (0.995 - 0.0950i)T \)
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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

−18.49332180766589873701774843263, −17.70728283032243151673033203135, −17.16951555168700342680013405065, −16.65959765212562365756271027206, −15.895455507133109406584820518048, −15.00406262943093194791199102360, −14.43949509854969481664143236299, −13.89486370629278589544412271948, −13.06021519657474336076790196352, −12.42456413460181637992486165690, −11.71182596359952758084170964234, −10.87557945092199200308850142171, −10.26261252913858106929053697228, −9.80065433172713245206260746955, −8.59218471134142895247771882467, −8.24489739411069971120002796300, −7.342887364262120968367615322934, −6.65533929090488701831878944141, −5.9584263747934967972009550120, −5.05259501088911317002998733255, −4.257197965633711551417314376910, −3.57112129719807060566792193555, −2.88811457537181326895813042521, −1.46826259202924900119612948461, −1.17186934969097970375708322046, 0.42893766534066000426094692693, 1.720331980772165440144520559572, 2.30251977223431195130074406722, 3.15550726449791767278004112315, 4.20923509785882775673818912033, 4.779588610014885557326424330234, 5.5919804539826234041221609084, 6.45938519603067172163931640080, 7.00167020438468934927318039191, 7.84100862052483346730248034277, 8.86012905413665500690567402974, 9.2046038858186683368360049346, 9.77505897623810473826923568425, 10.95936934810370281957639976531, 11.56292211323144381855716181784, 12.02587619878872739420419854838, 12.68198835606972082519150543303, 13.81419008938953098008693652420, 14.083177148338676511055488810276, 14.96135221090334305802603573658, 15.61071610122695426027317986786, 16.13434851110976604983562635952, 17.13785623468506919137323638800, 17.49943338973940808113551118077, 18.36318264451610880363331900369

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