Properties

Label 1-483-483.17-r1-0-0
Degree $1$
Conductor $483$
Sign $0.767 - 0.640i$
Analytic cond. $51.9055$
Root an. cond. $51.9055$
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 + 0.189i)2-s + (0.928 − 0.371i)4-s + (−0.580 − 0.814i)5-s + (−0.841 + 0.540i)8-s + (0.723 + 0.690i)10-s + (0.981 + 0.189i)11-s + (0.959 − 0.281i)13-s + (0.723 − 0.690i)16-s + (0.786 − 0.618i)17-s + (−0.786 − 0.618i)19-s + (−0.841 − 0.540i)20-s − 22-s + (−0.327 + 0.945i)25-s + (−0.888 + 0.458i)26-s + (0.142 + 0.989i)29-s + ⋯
L(s)  = 1  + (−0.981 + 0.189i)2-s + (0.928 − 0.371i)4-s + (−0.580 − 0.814i)5-s + (−0.841 + 0.540i)8-s + (0.723 + 0.690i)10-s + (0.981 + 0.189i)11-s + (0.959 − 0.281i)13-s + (0.723 − 0.690i)16-s + (0.786 − 0.618i)17-s + (−0.786 − 0.618i)19-s + (−0.841 − 0.540i)20-s − 22-s + (−0.327 + 0.945i)25-s + (−0.888 + 0.458i)26-s + (0.142 + 0.989i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.767 - 0.640i$
Analytic conductor: \(51.9055\)
Root analytic conductor: \(51.9055\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 483,\ (1:\ ),\ 0.767 - 0.640i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.199044522 - 0.4347870472i\)
\(L(\frac12)\) \(\approx\) \(1.199044522 - 0.4347870472i\)
\(L(1)\) \(\approx\) \(0.7598031181 - 0.09314635564i\)
\(L(1)\) \(\approx\) \(0.7598031181 - 0.09314635564i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.981 + 0.189i)T \)
5 \( 1 + (-0.580 - 0.814i)T \)
11 \( 1 + (0.981 + 0.189i)T \)
13 \( 1 + (0.959 - 0.281i)T \)
17 \( 1 + (0.786 - 0.618i)T \)
19 \( 1 + (-0.786 - 0.618i)T \)
29 \( 1 + (0.142 + 0.989i)T \)
31 \( 1 + (0.888 + 0.458i)T \)
37 \( 1 + (0.995 + 0.0950i)T \)
41 \( 1 + (0.415 - 0.909i)T \)
43 \( 1 + (-0.841 - 0.540i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.235 + 0.971i)T \)
59 \( 1 + (0.723 + 0.690i)T \)
61 \( 1 + (0.0475 + 0.998i)T \)
67 \( 1 + (0.327 - 0.945i)T \)
71 \( 1 + (0.654 + 0.755i)T \)
73 \( 1 + (-0.928 + 0.371i)T \)
79 \( 1 + (-0.235 + 0.971i)T \)
83 \( 1 + (-0.415 - 0.909i)T \)
89 \( 1 + (0.888 - 0.458i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
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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

−23.58896922284142975428529380533, −22.9070811738254034203645067045, −21.72902797656076035896905204808, −21.07055383704051598188888625102, −19.985229324965218001862202618413, −19.16209620517378772747163963195, −18.76620906967001802571980837095, −17.7842230434533705547842052416, −16.8348894081726753881681784523, −16.13799238947577087242418639298, −15.07583123697282774353825818036, −14.46294391900771836932882225112, −13.077565129295435966629589343299, −11.82525887885944236107635447006, −11.40808583320320654920561891661, −10.41006238568203940276312017060, −9.63006644902208622075545354326, −8.37877949489657781917780494204, −7.901257426869979047077890560989, −6.537901171730228533566781226405, −6.19575691613484723737540579831, −4.05266599438341691952305780972, −3.357423997792559942998284345112, −2.05105009002325561173178625430, −0.843672288370283421350613816253, 0.67836968908980907497606387040, 1.45890527516678271232667636397, 3.03510755205023335505705332868, 4.276846170437756087890734208034, 5.51354000839172148827440451543, 6.590072317037550996807798876548, 7.5261998819950561410472994031, 8.565236479930054142203537748464, 9.02144456292535804096371053936, 10.1123409500993312707287906028, 11.19232640626787147770979443645, 11.90125565032755647902586322414, 12.80242443618871223673437205762, 14.11680775330139976894750582794, 15.15113212350353961113783050704, 15.93647217886347585442691056859, 16.65352796698608456929226019613, 17.38396230172797202435342586803, 18.34355510021165108902374307391, 19.261526320754381941353875731925, 19.93636726773413857514142669940, 20.63892430526318519699160610788, 21.47797594603607341480243691869, 22.91144222977413302493336175002, 23.61670531577053130311125889030

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