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Properties

Label 2-480-24.11-c1-0-2
Degree $2$
Conductor $480$
Sign $0.644 - 0.764i$
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  + (−0.751 − 1.56i)3-s + 5-s + 4.28i·7-s + (−1.86 + 2.34i)9-s + 2.44i·11-s + 2.71i·13-s + (−0.751 − 1.56i)15-s + 1.16i·17-s − 6.05·19-s + (6.68 − 3.22i)21-s + 7.55·23-s + 25-s + (5.06 + 1.15i)27-s + 0.733·29-s + 0.469i·31-s + ⋯
L(s)  = 1  + (−0.433 − 0.900i)3-s + 0.447·5-s + 1.61i·7-s + (−0.623 + 0.781i)9-s + 0.737i·11-s + 0.752i·13-s + (−0.194 − 0.402i)15-s + 0.282i·17-s − 1.38·19-s + (1.45 − 0.703i)21-s + 1.57·23-s + 0.200·25-s + (0.975 + 0.222i)27-s + 0.136·29-s + 0.0843i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(480\)    =    \(2^{5} \cdot 3 \cdot 5\)
Sign: $0.644 - 0.764i$
Analytic conductor: \(3.83281\)
Root analytic conductor: \(1.95775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{480} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 480,\ (\ :1/2),\ 0.644 - 0.764i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03143 + 0.479576i\)
\(L(\frac12)\) \(\approx\) \(1.03143 + 0.479576i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.751 + 1.56i)T \)
5 \( 1 - T \)
good7 \( 1 - 4.28iT - 7T^{2} \)
11 \( 1 - 2.44iT - 11T^{2} \)
13 \( 1 - 2.71iT - 13T^{2} \)
17 \( 1 - 1.16iT - 17T^{2} \)
19 \( 1 + 6.05T + 19T^{2} \)
23 \( 1 - 7.55T + 23T^{2} \)
29 \( 1 - 0.733T + 29T^{2} \)
31 \( 1 - 0.469iT - 31T^{2} \)
37 \( 1 - 1.36iT - 37T^{2} \)
41 \( 1 + 4.69iT - 41T^{2} \)
43 \( 1 - 1.50T + 43T^{2} \)
47 \( 1 - 4.07T + 47T^{2} \)
53 \( 1 - 1.00T + 53T^{2} \)
59 \( 1 + 1.63iT - 59T^{2} \)
61 \( 1 - 10.9iT - 61T^{2} \)
67 \( 1 - 9.97T + 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + 9.63T + 73T^{2} \)
79 \( 1 + 3.61iT - 79T^{2} \)
83 \( 1 + 5.45iT - 83T^{2} \)
89 \( 1 + 7.75iT - 89T^{2} \)
97 \( 1 - 17.1T + 97T^{2} \)
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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

−11.31147583181028579548150158273, −10.33215620191168336043270068330, −9.016302200869418798254627398982, −8.642963177305129783041104414319, −7.27827381942074464845581831719, −6.42782457464440950124337847127, −5.65771931955990958013172331684, −4.67770805864401943511097100836, −2.63095688679300040872473340185, −1.80510709614391523757876563442, 0.74594056837735731427249108274, 3.07301379258703542300534998112, 4.11775574275003069359570439052, 5.04953146476661070575566208663, 6.15201828121621944795778279250, 7.05569967445951754007186960919, 8.283683674953069799068260202488, 9.254026787459650618844650810070, 10.24057392596262651046329887853, 10.74129313562626529514124108954

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