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Properties

Label 1-1520-1520.883-r0-0-0
Degree $1$
Conductor $1520$
Sign $0.929 + 0.369i$
Analytic cond. $7.05885$
Root an. cond. $7.05885$
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 + (−0.866 + 0.5i)7-s + (−0.939 − 0.342i)9-s + (−0.866 − 0.5i)11-s + (−0.173 − 0.984i)13-s + (0.342 + 0.939i)17-s + (0.342 + 0.939i)21-s + (0.642 + 0.766i)23-s + (−0.5 + 0.866i)27-s + (−0.342 + 0.939i)29-s + (0.5 + 0.866i)31-s + (−0.642 + 0.766i)33-s − 37-s − 39-s + (−0.173 + 0.984i)41-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)3-s + (−0.866 + 0.5i)7-s + (−0.939 − 0.342i)9-s + (−0.866 − 0.5i)11-s + (−0.173 − 0.984i)13-s + (0.342 + 0.939i)17-s + (0.342 + 0.939i)21-s + (0.642 + 0.766i)23-s + (−0.5 + 0.866i)27-s + (−0.342 + 0.939i)29-s + (0.5 + 0.866i)31-s + (−0.642 + 0.766i)33-s − 37-s − 39-s + (−0.173 + 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.929 + 0.369i$
Analytic conductor: \(7.05885\)
Root analytic conductor: \(7.05885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1520,\ (0:\ ),\ 0.929 + 0.369i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9189077311 + 0.1760234884i\)
\(L(\frac12)\) \(\approx\) \(0.9189077311 + 0.1760234884i\)
\(L(1)\) \(\approx\) \(0.8551525953 - 0.1699284477i\)
\(L(1)\) \(\approx\) \(0.8551525953 - 0.1699284477i\)

Euler product

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

−20.78455621505991014430547799126, −19.94422715271976948924333450343, −19.06936623952208169206877998526, −18.51598879867872174199863724, −17.158294619710272842230426375240, −16.854974805733531811391109198707, −15.85716575034248806769400593010, −15.580474300301671568397727019, −14.503676404670044559727437279547, −13.850522053547148799120590369916, −13.09571977204651553444563141832, −12.12498826607357601854535148352, −11.258090650254116346559898262542, −10.44216176091421895843435635278, −9.73802153728754646929641573769, −9.29740773319033748354236524282, −8.24332298993053178650833320706, −7.30977231266675701318855834631, −6.50941573587430872287617787937, −5.42610292844399740100812139994, −4.6407459216983520037229478313, −3.91465647710564238875708085869, −2.95283181044007319515793338130, −2.21772347730065100979996010399, −0.40854756073477489760475628946, 0.9160335816391059116120908915, 2.06376224004396556963709667375, 3.06997673463518076316286760549, 3.45807919659680394467895780328, 5.31961794559565976648717891589, 5.640755863711767745962829773165, 6.68657895895993208298957247101, 7.35387502814993264205038035834, 8.37581154043961797827096726598, 8.7369756054819557939892318778, 9.98825638815278965895044434806, 10.63140792563752972690003817537, 11.7235292596474978665107981851, 12.44758898235040715411718292767, 13.11939122724212446503013342395, 13.486536001306223188336669598971, 14.679140064275654563912974365688, 15.28489326086403401715764017543, 16.13959136483247179973160736570, 17.007579733628651148432424685754, 17.812446195564758145312135187103, 18.471176128948538541784879376093, 19.17150107550862843900997485250, 19.66690592431994988472644991017, 20.480372910384768778277042810142

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