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Properties

Label 2-580-580.499-c0-0-2
Degree $2$
Conductor $580$
Sign $0.990 + 0.140i$
Analytic cond. $0.289457$
Root an. cond. $0.538012$
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  + (0.781 + 0.623i)2-s + (−1.21 − 0.277i)3-s + (0.222 + 0.974i)4-s + (0.623 − 0.781i)5-s + (−0.777 − 0.974i)6-s + (0.433 − 1.90i)7-s + (−0.433 + 0.900i)8-s + (0.499 + 0.240i)9-s + (0.974 − 0.222i)10-s − 1.24i·12-s + (1.52 − 1.21i)14-s + (−0.974 + 0.777i)15-s + (−0.900 + 0.433i)16-s + (0.240 + 0.5i)18-s + (0.900 + 0.433i)20-s + (−1.05 + 2.19i)21-s + ⋯
L(s)  = 1  + (0.781 + 0.623i)2-s + (−1.21 − 0.277i)3-s + (0.222 + 0.974i)4-s + (0.623 − 0.781i)5-s + (−0.777 − 0.974i)6-s + (0.433 − 1.90i)7-s + (−0.433 + 0.900i)8-s + (0.499 + 0.240i)9-s + (0.974 − 0.222i)10-s − 1.24i·12-s + (1.52 − 1.21i)14-s + (−0.974 + 0.777i)15-s + (−0.900 + 0.433i)16-s + (0.240 + 0.5i)18-s + (0.900 + 0.433i)20-s + (−1.05 + 2.19i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(580\)    =    \(2^{2} \cdot 5 \cdot 29\)
Sign: $0.990 + 0.140i$
Analytic conductor: \(0.289457\)
Root analytic conductor: \(0.538012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{580} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 580,\ (\ :0),\ 0.990 + 0.140i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.045428213\)
\(L(\frac12)\) \(\approx\) \(1.045428213\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.781 - 0.623i)T \)
5 \( 1 + (-0.623 + 0.781i)T \)
29 \( 1 + (0.222 - 0.974i)T \)
good3 \( 1 + (1.21 + 0.277i)T + (0.900 + 0.433i)T^{2} \)
7 \( 1 + (-0.433 + 1.90i)T + (-0.900 - 0.433i)T^{2} \)
11 \( 1 + (0.623 - 0.781i)T^{2} \)
13 \( 1 + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-0.900 + 0.433i)T^{2} \)
23 \( 1 + (-0.974 - 1.22i)T + (-0.222 + 0.974i)T^{2} \)
31 \( 1 + (-0.222 - 0.974i)T^{2} \)
37 \( 1 + (0.623 + 0.781i)T^{2} \)
41 \( 1 + 0.867iT - T^{2} \)
43 \( 1 + (1.40 - 1.12i)T + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (-0.193 - 0.400i)T + (-0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.222 + 0.974i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.900 + 0.433i)T^{2} \)
67 \( 1 + (0.623 + 0.781i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.222 + 0.974i)T^{2} \)
79 \( 1 + (0.623 + 0.781i)T^{2} \)
83 \( 1 + (0.193 + 0.846i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (-1.52 - 1.21i)T + (0.222 + 0.974i)T^{2} \)
97 \( 1 + (-0.900 + 0.433i)T^{2} \)
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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

−11.10440857233683616721814715681, −10.32722851010996678843793030831, −9.068008264534020139642808482772, −7.85072769911313247365909372963, −7.07512642404202687916153560404, −6.33038342799783321927213605971, −5.23637073747763124049328478596, −4.77164184119076722042817555167, −3.59158873308908596442473987257, −1.31532271764520104370343220137, 2.06680096340152300433337871802, 2.97941860969455483416574601799, 4.70298792142108900848169053782, 5.43989318379741721534906162133, 6.03251503774890426343301881366, 6.75120795074519170810508909939, 8.558016499413008721910057097391, 9.563556702542345922284446868664, 10.39215439816425411735005295160, 11.17865547387548250379039166642

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