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Properties

Label 1-1480-1480.123-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.996 - 0.0784i$
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  + (−0.984 + 0.173i)3-s + (0.642 + 0.766i)7-s + (0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.342 + 0.939i)13-s + (0.342 + 0.939i)17-s + (−0.173 − 0.984i)19-s + (−0.766 − 0.642i)21-s + (−0.866 + 0.5i)23-s + (−0.866 + 0.5i)27-s + (−0.5 + 0.866i)29-s − 31-s + (0.342 − 0.939i)33-s + (0.173 − 0.984i)39-s + (−0.939 − 0.342i)41-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)3-s + (0.642 + 0.766i)7-s + (0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.342 + 0.939i)13-s + (0.342 + 0.939i)17-s + (−0.173 − 0.984i)19-s + (−0.766 − 0.642i)21-s + (−0.866 + 0.5i)23-s + (−0.866 + 0.5i)27-s + (−0.5 + 0.866i)29-s − 31-s + (0.342 − 0.939i)33-s + (0.173 − 0.984i)39-s + (−0.939 − 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.996 - 0.0784i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.996 - 0.0784i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01875806128 + 0.4774811751i\)
\(L(\frac12)\) \(\approx\) \(0.01875806128 + 0.4774811751i\)
\(L(1)\) \(\approx\) \(0.6474278692 + 0.2439544307i\)
\(L(1)\) \(\approx\) \(0.6474278692 + 0.2439544307i\)

Euler product

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

−20.54820124007241300971068405745, −19.44773940584612532368035667102, −18.37459753959973892737183164637, −18.24378734166656275465631925099, −17.11068390738669247348384453177, −16.71029278139783301383409364098, −15.952366433921629179772542056178, −15.06821490653787037377425473236, −14.04445536579113311916673368966, −13.46744437324733280630861741074, −12.536358537498386754771620197897, −11.84142834506520158052509348689, −11.021536902507696707570413058674, −10.42623621299346127387790296181, −9.80782831162594013141057291441, −8.38458977154856546324393063430, −7.69406890157231965096538094907, −7.0774232083045744844357860469, −5.82568003810040487086601923778, −5.47264308458956711729591092764, −4.45932750372195453867545591087, −3.60217481256395334567201600801, −2.32469468571225191442622089098, −1.16434781159124319689343621204, −0.21844958108713030469818367009, 1.588195018878072352364653613, 2.15799046370245356569878555307, 3.672139479828835435699204630298, 4.63347210529003384332475048940, 5.20306792903605918258330445208, 6.00967463574079959908182978019, 6.95384570060732781115678364401, 7.66292241230689439001085511493, 8.7913812274416134150104972730, 9.55722077538606176559456290214, 10.40479560944948767629205711581, 11.176634528093090306963306707599, 11.88293309748307703240530239996, 12.48172822728976422716738761330, 13.24094783516373068616894103575, 14.47835549634780466027393189481, 15.09578382005596485897190497705, 15.76517142329741139029978868885, 16.62973308315963621797000937212, 17.34259564075780073427833103968, 18.02681534638878873700662920268, 18.53110931520646765627851079150, 19.471267238209118402516323670834, 20.41418372484811105516895729708, 21.33434477211982280583500625982

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