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Properties

Label 2-480-120.59-c1-0-18
Degree $2$
Conductor $480$
Sign $-0.866 + 0.499i$
Analytic cond. $3.83281$
Root an. cond. $1.95775$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.541 + 1.64i)3-s + (−1.25 − 1.84i)5-s − 3.29·7-s + (−2.41 + 1.78i)9-s − 2.51i·11-s − 4.65·13-s + (2.35 − 3.07i)15-s − 3.69·17-s − 0.828·19-s + (−1.78 − 5.41i)21-s + 2.61i·23-s + (−1.82 + 4.65i)25-s + (−4.23 − 3.00i)27-s + 6.08·29-s − 1.17i·31-s + ⋯
L(s)  = 1  + (0.312 + 0.949i)3-s + (−0.563 − 0.826i)5-s − 1.24·7-s + (−0.804 + 0.593i)9-s − 0.759i·11-s − 1.29·13-s + (0.609 − 0.793i)15-s − 0.896·17-s − 0.190·19-s + (−0.388 − 1.18i)21-s + 0.544i·23-s + (−0.365 + 0.930i)25-s + (−0.815 − 0.578i)27-s + 1.12·29-s − 0.210i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(480\)    =    \(2^{5} \cdot 3 \cdot 5\)
Sign: $-0.866 + 0.499i$
Analytic conductor: \(3.83281\)
Root analytic conductor: \(1.95775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{480} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 480,\ (\ :1/2),\ -0.866 + 0.499i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0173744 - 0.0648815i\)
\(L(\frac12)\) \(\approx\) \(0.0173744 - 0.0648815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.541 - 1.64i)T \)
5 \( 1 + (1.25 + 1.84i)T \)
good7 \( 1 + 3.29T + 7T^{2} \)
11 \( 1 + 2.51iT - 11T^{2} \)
13 \( 1 + 4.65T + 13T^{2} \)
17 \( 1 + 3.69T + 17T^{2} \)
19 \( 1 + 0.828T + 19T^{2} \)
23 \( 1 - 2.61iT - 23T^{2} \)
29 \( 1 - 6.08T + 29T^{2} \)
31 \( 1 + 1.17iT - 31T^{2} \)
37 \( 1 + 1.92T + 37T^{2} \)
41 \( 1 + 8.59iT - 41T^{2} \)
43 \( 1 - 6.01iT - 43T^{2} \)
47 \( 1 + 2.61iT - 47T^{2} \)
53 \( 1 + 4.59iT - 53T^{2} \)
59 \( 1 - 2.51iT - 59T^{2} \)
61 \( 1 - 8.48iT - 61T^{2} \)
67 \( 1 + 3.29iT - 67T^{2} \)
71 \( 1 + 7.12T + 71T^{2} \)
73 \( 1 + 6.58iT - 73T^{2} \)
79 \( 1 - 16.4iT - 79T^{2} \)
83 \( 1 + 9.37T + 83T^{2} \)
89 \( 1 - 5.03iT - 89T^{2} \)
97 \( 1 - 2.72iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.45394047295067281288109194764, −9.619855224811675126337806790602, −8.974183766632125500670030268166, −8.168598834390323976057172095660, −6.97704348593018942491172366455, −5.71021782446190454005226029393, −4.71284630556357275709714077327, −3.76677137715374749172955761093, −2.70943155127400307901055441939, −0.03598072663752730332199704156, 2.35437601837322649655545970347, 3.14442197958374586761624721747, 4.54219611675866452836130354221, 6.25094086468892190587100224204, 6.85969774804129167551762153445, 7.47122189964934202438616711932, 8.564892667112374209662008406804, 9.632037196357456112018785006948, 10.38551169155033782404740794961, 11.56289756431082395811209772022

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