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Properties

Label 2-720-60.59-c1-0-5
Degree $2$
Conductor $720$
Sign $0.898 - 0.438i$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
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  + (1.73 + 1.41i)5-s + 4.24·7-s + 2.44·11-s + 2.44i·13-s − 6.92·17-s − 6.92i·19-s + 6i·23-s + (0.999 + 4.89i)25-s − 2.82i·29-s − 3.46i·31-s + (7.34 + 6i)35-s − 2.44i·37-s + 7.07i·41-s + 8.48·43-s + 10.9·49-s + ⋯
L(s)  = 1  + (0.774 + 0.632i)5-s + 1.60·7-s + 0.738·11-s + 0.679i·13-s − 1.68·17-s − 1.58i·19-s + 1.25i·23-s + (0.199 + 0.979i)25-s − 0.525i·29-s − 0.622i·31-s + (1.24 + 1.01i)35-s − 0.402i·37-s + 1.10i·41-s + 1.29·43-s + 1.57·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.898 - 0.438i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.898 - 0.438i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98890 + 0.459775i\)
\(L(\frac12)\) \(\approx\) \(1.98890 + 0.459775i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-1.73 - 1.41i)T \)
good7 \( 1 - 4.24T + 7T^{2} \)
11 \( 1 - 2.44T + 11T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + 6.92T + 17T^{2} \)
19 \( 1 + 6.92iT - 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + 2.44iT - 37T^{2} \)
41 \( 1 - 7.07iT - 41T^{2} \)
43 \( 1 - 8.48T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 9.79T + 71T^{2} \)
73 \( 1 - 4.89iT - 73T^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 7.07iT - 89T^{2} \)
97 \( 1 - 14.6iT - 97T^{2} \)
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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

−10.74144142356281780731504214161, −9.325928920097388352664160552063, −9.058356613524851359310684766163, −7.78958875895327564852502006037, −6.94138167883631523322377738023, −6.12154345302736998317516409372, −4.94424341309769903900660912350, −4.18983299829703950659083867091, −2.51141642770941119066743926451, −1.60862282540554099598551133693, 1.31244655177094474016514722907, 2.27296326899149391535151945752, 4.11316086485883064950339852275, 4.88164305166889558274535286802, 5.77784284036609080307433316310, 6.75213934026362443293800521710, 8.035044458421903837424820348285, 8.576461665859951066702543791458, 9.343482838990034842327880929618, 10.55282045705349167772968681551

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