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Properties

Label 1-1480-1480.589-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.806 - 0.590i$
Analytic cond. $6.87309$
Root an. cond. $6.87309$
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  + (0.173 − 0.984i)3-s + (−0.766 − 0.642i)7-s + (−0.939 − 0.342i)9-s + (0.5 + 0.866i)11-s + (−0.939 + 0.342i)13-s + (0.939 + 0.342i)17-s + (−0.173 + 0.984i)19-s + (−0.766 + 0.642i)21-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)27-s + (0.5 + 0.866i)29-s + 31-s + (0.939 − 0.342i)33-s + (0.173 + 0.984i)39-s + (−0.939 + 0.342i)41-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)3-s + (−0.766 − 0.642i)7-s + (−0.939 − 0.342i)9-s + (0.5 + 0.866i)11-s + (−0.939 + 0.342i)13-s + (0.939 + 0.342i)17-s + (−0.173 + 0.984i)19-s + (−0.766 + 0.642i)21-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)27-s + (0.5 + 0.866i)29-s + 31-s + (0.939 − 0.342i)33-s + (0.173 + 0.984i)39-s + (−0.939 + 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.806 - 0.590i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.806 - 0.590i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.321340704 - 0.4320969400i\)
\(L(\frac12)\) \(\approx\) \(1.321340704 - 0.4320969400i\)
\(L(1)\) \(\approx\) \(0.9973743282 - 0.2783817577i\)
\(L(1)\) \(\approx\) \(0.9973743282 - 0.2783817577i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (0.173 - 0.984i)T \)
7 \( 1 + (-0.766 - 0.642i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 - T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (-0.939 - 0.342i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−20.90892337697265981803125359433, −19.87057309587660954039734087470, −19.33186571031238361877168289475, −18.78345703578529432752380987187, −17.339933311831803071520501006362, −17.07626529841149489474235279245, −15.98770284442154230984541076155, −15.605175554346280181242278204195, −14.819770916053011197215688389159, −14.00496620727002636282587788843, −13.28297257824728778268443209945, −12.14098230704516777843528885939, −11.61781830797952908124529643486, −10.64650455856936416421806194646, −9.80033485352736322215044878181, −9.29733848429134384434172082617, −8.55497244489878603310604050921, −7.59832427451091960500557917069, −6.46897770690095448360066296373, −5.631647330670867309032647347168, −4.97332702634380921802672534249, −3.908349149674198026853331075484, −3.01463850871321852304400106838, −2.53730396438396324225390114282, −0.71209303009260246132379236407, 0.83374815736246160108739122897, 1.80519519513321147494284279523, 2.78342707025249555894169263906, 3.68620961807320234867417119755, 4.68719986510552854026034113, 5.82938914957827900236945563757, 6.74466876855969400463096115737, 7.13448218255977515525070695694, 8.01434020854240499894595744494, 8.90981867340367696923463477099, 9.89883856415878685402288512824, 10.398091377686154997170260075730, 11.73810909941435483443313695537, 12.3828725410478264467617527149, 12.76663994496896042003813059228, 13.814649653206281593145104913604, 14.43911204902008993674605202060, 15.02063892021902861643853757535, 16.34761218554412526518138373091, 16.9589436896783602432370629283, 17.46548027818292236973628305141, 18.54373459738015602539286860455, 19.0883117278501093285170190681, 19.77359526035069931225120868644, 20.33036445160633389408160778736

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