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Properties

Label 2-580-580.419-c0-0-0
Degree $2$
Conductor $580$
Sign $-0.776 - 0.629i$
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  + (−0.433 + 0.900i)2-s + (1.40 + 1.12i)3-s + (−0.623 − 0.781i)4-s + (−0.900 − 0.433i)5-s + (−1.62 + 0.781i)6-s + (−0.974 + 1.22i)7-s + (0.974 − 0.222i)8-s + (0.500 + 2.19i)9-s + (0.781 − 0.623i)10-s − 1.80i·12-s + (−0.678 − 1.40i)14-s + (−0.781 − 1.62i)15-s + (−0.222 + 0.974i)16-s + (−2.19 − 0.499i)18-s + (0.222 + 0.974i)20-s + (−2.74 + 0.626i)21-s + ⋯
L(s)  = 1  + (−0.433 + 0.900i)2-s + (1.40 + 1.12i)3-s + (−0.623 − 0.781i)4-s + (−0.900 − 0.433i)5-s + (−1.62 + 0.781i)6-s + (−0.974 + 1.22i)7-s + (0.974 − 0.222i)8-s + (0.500 + 2.19i)9-s + (0.781 − 0.623i)10-s − 1.80i·12-s + (−0.678 − 1.40i)14-s + (−0.781 − 1.62i)15-s + (−0.222 + 0.974i)16-s + (−2.19 − 0.499i)18-s + (0.222 + 0.974i)20-s + (−2.74 + 0.626i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(580\)    =    \(2^{2} \cdot 5 \cdot 29\)
Sign: $-0.776 - 0.629i$
Analytic conductor: \(0.289457\)
Root analytic conductor: \(0.538012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{580} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 580,\ (\ :0),\ -0.776 - 0.629i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8540164826\)
\(L(\frac12)\) \(\approx\) \(0.8540164826\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.433 - 0.900i)T \)
5 \( 1 + (0.900 + 0.433i)T \)
29 \( 1 + (-0.623 + 0.781i)T \)
good3 \( 1 + (-1.40 - 1.12i)T + (0.222 + 0.974i)T^{2} \)
7 \( 1 + (0.974 - 1.22i)T + (-0.222 - 0.974i)T^{2} \)
11 \( 1 + (-0.900 - 0.433i)T^{2} \)
13 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-0.222 + 0.974i)T^{2} \)
23 \( 1 + (-0.781 + 0.376i)T + (0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.623 + 0.781i)T^{2} \)
37 \( 1 + (-0.900 + 0.433i)T^{2} \)
41 \( 1 + 1.94iT - T^{2} \)
43 \( 1 + (-0.193 - 0.400i)T + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (-1.21 - 0.277i)T + (0.900 + 0.433i)T^{2} \)
53 \( 1 + (-0.623 - 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.222 + 0.974i)T^{2} \)
67 \( 1 + (-0.900 + 0.433i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.623 - 0.781i)T^{2} \)
79 \( 1 + (-0.900 + 0.433i)T^{2} \)
83 \( 1 + (1.21 + 1.52i)T + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.678 - 1.40i)T + (-0.623 - 0.781i)T^{2} \)
97 \( 1 + (-0.222 + 0.974i)T^{2} \)
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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

−10.85812007296629745645410005522, −9.943195330128548406262385731854, −9.130418484682206414688814322663, −8.801409251595727862339071335694, −8.065965683705727013593712801940, −7.08015034892948035942476230428, −5.70664640604928630725887829447, −4.68811173550812426024995626648, −3.76204784907551265284541038154, −2.60935201895516516126512201721, 1.10609730403724994252104050984, 2.78687499029036030319888381172, 3.37875345614187953209404857333, 4.23820192031684352830138451029, 6.75247819181205555726327199917, 7.23784695371157476650523238589, 7.969235452785484506602032316331, 8.772372768609732000161087150582, 9.659827825338161473298331307879, 10.50654822177711161026543718353

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