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Properties

Label 2-580-580.419-c0-0-2
Degree $2$
Conductor $580$
Sign $-0.00819 + 0.999i$
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  + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (0.222 − 0.974i)5-s + (−0.222 − 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.623 + 0.781i)10-s + (−1.52 − 0.347i)13-s + (−0.222 + 0.974i)16-s + 1.80·17-s + (−0.222 + 0.974i)18-s + (0.900 − 0.433i)20-s + (−0.900 − 0.433i)25-s + (1.22 + 0.974i)26-s + (0.222 − 0.974i)29-s + (0.623 − 0.781i)32-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (0.222 − 0.974i)5-s + (−0.222 − 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.623 + 0.781i)10-s + (−1.52 − 0.347i)13-s + (−0.222 + 0.974i)16-s + 1.80·17-s + (−0.222 + 0.974i)18-s + (0.900 − 0.433i)20-s + (−0.900 − 0.433i)25-s + (1.22 + 0.974i)26-s + (0.222 − 0.974i)29-s + (0.623 − 0.781i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00819 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00819 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(580\)    =    \(2^{2} \cdot 5 \cdot 29\)
Sign: $-0.00819 + 0.999i$
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: \((2,\ 580,\ (\ :0),\ -0.00819 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5916711086\)
\(L(\frac12)\) \(\approx\) \(0.5916711086\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.900 + 0.433i)T \)
5 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (-0.222 + 0.974i)T \)
good3 \( 1 + (0.222 + 0.974i)T^{2} \)
7 \( 1 + (-0.222 - 0.974i)T^{2} \)
11 \( 1 + (-0.900 - 0.433i)T^{2} \)
13 \( 1 + (1.52 + 0.347i)T + (0.900 + 0.433i)T^{2} \)
17 \( 1 - 1.80T + T^{2} \)
19 \( 1 + (-0.222 + 0.974i)T^{2} \)
23 \( 1 + (0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.623 + 0.781i)T^{2} \)
37 \( 1 + (-0.277 - 1.21i)T + (-0.900 + 0.433i)T^{2} \)
41 \( 1 + 1.56iT - T^{2} \)
43 \( 1 + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.900 + 0.433i)T^{2} \)
53 \( 1 + (0.376 - 0.781i)T + (-0.623 - 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.678 - 0.541i)T + (0.222 + 0.974i)T^{2} \)
67 \( 1 + (-0.900 + 0.433i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \)
79 \( 1 + (-0.900 + 0.433i)T^{2} \)
83 \( 1 + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.846 - 1.75i)T + (-0.623 - 0.781i)T^{2} \)
97 \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \)
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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

−10.38560838895886154265253898632, −9.710193262600687845961269645002, −9.202172644692105682967168649472, −8.127608055746079847358013667722, −7.50202937962206203170318903768, −6.24545276788656037600926472239, −5.17128142779901383393917760622, −3.80899876621016052401476391686, −2.58036098374940259624040257295, −0.974021854137760207987405661264, 1.99511065622949100237121090086, 3.04840576461301472614308537612, 4.99688619289435750714136435077, 5.79517032363286600231751767790, 6.99456366694300532253474970825, 7.52632301776874556676190379813, 8.337621867457639573405292453998, 9.726693322343953261540183537096, 9.988917598154381663673301888891, 10.93595239171206363730687515435

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