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Properties

Label 1-4024-4024.1227-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.0734 + 0.997i$
Analytic cond. $18.6873$
Root an. cond. $18.6873$
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  + (−0.325 − 0.945i)3-s + (0.289 + 0.957i)5-s + (0.986 − 0.161i)7-s + (−0.787 + 0.615i)9-s + (−0.977 − 0.211i)11-s + (−0.337 − 0.941i)13-s + (0.810 − 0.585i)15-s + (0.982 + 0.186i)17-s + (−0.560 − 0.828i)19-s + (−0.474 − 0.880i)21-s + (−0.731 − 0.682i)23-s + (−0.831 + 0.554i)25-s + (0.838 + 0.544i)27-s + (−0.205 − 0.978i)29-s + (−0.301 + 0.953i)31-s + ⋯
L(s)  = 1  + (−0.325 − 0.945i)3-s + (0.289 + 0.957i)5-s + (0.986 − 0.161i)7-s + (−0.787 + 0.615i)9-s + (−0.977 − 0.211i)11-s + (−0.337 − 0.941i)13-s + (0.810 − 0.585i)15-s + (0.982 + 0.186i)17-s + (−0.560 − 0.828i)19-s + (−0.474 − 0.880i)21-s + (−0.731 − 0.682i)23-s + (−0.831 + 0.554i)25-s + (0.838 + 0.544i)27-s + (−0.205 − 0.978i)29-s + (−0.301 + 0.953i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0734 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0734 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.0734 + 0.997i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.0734 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5080217526 + 0.4719953107i\)
\(L(\frac12)\) \(\approx\) \(0.5080217526 + 0.4719953107i\)
\(L(1)\) \(\approx\) \(0.8619645718 - 0.1051430958i\)
\(L(1)\) \(\approx\) \(0.8619645718 - 0.1051430958i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.325 - 0.945i)T \)
5 \( 1 + (0.289 + 0.957i)T \)
7 \( 1 + (0.986 - 0.161i)T \)
11 \( 1 + (-0.977 - 0.211i)T \)
13 \( 1 + (-0.337 - 0.941i)T \)
17 \( 1 + (0.982 + 0.186i)T \)
19 \( 1 + (-0.560 - 0.828i)T \)
23 \( 1 + (-0.731 - 0.682i)T \)
29 \( 1 + (-0.205 - 0.978i)T \)
31 \( 1 + (-0.301 + 0.953i)T \)
37 \( 1 + (-0.739 + 0.673i)T \)
41 \( 1 + (0.00625 + 0.999i)T \)
43 \( 1 + (-0.845 + 0.533i)T \)
47 \( 1 + (-0.994 + 0.0999i)T \)
53 \( 1 + (0.560 - 0.828i)T \)
59 \( 1 + (0.496 + 0.868i)T \)
61 \( 1 + (-0.118 + 0.992i)T \)
67 \( 1 + (0.810 + 0.585i)T \)
71 \( 1 + (-0.106 + 0.994i)T \)
73 \( 1 + (-0.549 - 0.835i)T \)
79 \( 1 + (-0.988 + 0.149i)T \)
83 \( 1 + (-0.971 + 0.235i)T \)
89 \( 1 + (0.934 + 0.355i)T \)
97 \( 1 + (0.0937 + 0.995i)T \)
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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

−18.26857096801442906989779542988, −17.29815229935481644934607442175, −17.05023491621395495297531100019, −16.2226849206451475937787478225, −15.80617423790827782064825256080, −14.88731667432558441356242900347, −14.33133737094543134762051932731, −13.70184970893627501721649045891, −12.61378809436761681831856552387, −12.09857188988339433517802653147, −11.474843910815575218675401239929, −10.64448136678910576613821554312, −9.97815151958210985527878404441, −9.39657114380867656085610997020, −8.59987311420654678604965544979, −8.04975750773125896286214113501, −7.20907899785070929441750864965, −5.91543537706579554930502451499, −5.41413051696679936870957286603, −4.91717703044752516784819829171, −4.18147066292476143610606709154, −3.4612209765909645208354562656, −2.12662612669112060516243690952, −1.627390215981746253532293745830, −0.20838094164318410065228892692, 1.009527822321486888358990518037, 1.97289607658328933782706289731, 2.63015734191800864953181757069, 3.28599508145505764086976749659, 4.61929443264578130577080357456, 5.38146073994082513300877285300, 5.913245694791762970253235798855, 6.80899337849412679160950803193, 7.42062940211584335092001730373, 8.13385686987719389303434837752, 8.460800413936490984103915673, 10.10593573663179051347041450835, 10.295155736268599092445091644077, 11.18503840927676425576073612101, 11.64457657963828733945806418087, 12.524014089062300004287800867486, 13.26201598692302401871049960860, 13.733598370900052941171333061173, 14.73084514740435729401500084199, 14.8270623877898772343387101273, 15.94431254871180005683152798001, 16.83000357082937620776779365359, 17.57624043012056882936368115531, 17.95851303417569619550543465272, 18.451137675804149216543428730846

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