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Properties

Label 2-580-580.427-c0-0-0
Degree $2$
Conductor $580$
Sign $0.997 + 0.0676i$
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 + (0.974 − 0.222i)5-s + (−0.222 + 0.974i)8-s + (0.222 − 0.974i)9-s + (−0.781 + 0.623i)10-s + (0.119 − 0.189i)13-s + (−0.222 − 0.974i)16-s − 0.867·17-s + (0.222 + 0.974i)18-s + (0.433 − 0.900i)20-s + (0.900 − 0.433i)25-s + (−0.0250 + 0.222i)26-s + (−0.974 + 0.222i)29-s + (0.623 + 0.781i)32-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (0.974 − 0.222i)5-s + (−0.222 + 0.974i)8-s + (0.222 − 0.974i)9-s + (−0.781 + 0.623i)10-s + (0.119 − 0.189i)13-s + (−0.222 − 0.974i)16-s − 0.867·17-s + (0.222 + 0.974i)18-s + (0.433 − 0.900i)20-s + (0.900 − 0.433i)25-s + (−0.0250 + 0.222i)26-s + (−0.974 + 0.222i)29-s + (0.623 + 0.781i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(580\)    =    \(2^{2} \cdot 5 \cdot 29\)
Sign: $0.997 + 0.0676i$
Analytic conductor: \(0.289457\)
Root analytic conductor: \(0.538012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{580} (427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 580,\ (\ :0),\ 0.997 + 0.0676i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7181160618\)
\(L(\frac12)\) \(\approx\) \(0.7181160618\)
\(L(1)\) 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 \( 1 + (-0.974 + 0.222i)T \)
29 \( 1 + (0.974 - 0.222i)T \)
good3 \( 1 + (-0.222 + 0.974i)T^{2} \)
7 \( 1 + (0.974 + 0.222i)T^{2} \)
11 \( 1 + (-0.433 - 0.900i)T^{2} \)
13 \( 1 + (-0.119 + 0.189i)T + (-0.433 - 0.900i)T^{2} \)
17 \( 1 + 0.867T + T^{2} \)
19 \( 1 + (-0.974 + 0.222i)T^{2} \)
23 \( 1 + (-0.781 + 0.623i)T^{2} \)
31 \( 1 + (0.781 + 0.623i)T^{2} \)
37 \( 1 + (-1.21 - 0.277i)T + (0.900 + 0.433i)T^{2} \)
41 \( 1 + (-1.40 - 1.40i)T + iT^{2} \)
43 \( 1 + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (-0.900 + 0.433i)T^{2} \)
53 \( 1 + (0.623 - 1.78i)T + (-0.781 - 0.623i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (-0.656 - 0.0739i)T + (0.974 + 0.222i)T^{2} \)
67 \( 1 + (0.433 - 0.900i)T^{2} \)
71 \( 1 + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (1.62 + 0.781i)T + (0.623 + 0.781i)T^{2} \)
79 \( 1 + (-0.433 + 0.900i)T^{2} \)
83 \( 1 + (0.974 - 0.222i)T^{2} \)
89 \( 1 + (1.00 + 0.351i)T + (0.781 + 0.623i)T^{2} \)
97 \( 1 + (1.22 + 0.974i)T + (0.222 + 0.974i)T^{2} \)
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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

−10.73562660683946360675082117585, −9.673054482159271619572933741290, −9.338953811063912605576300602229, −8.448368396388228754427793266674, −7.34653706057705363950325735902, −6.36442361073689490449061665002, −5.83578256167267737171192133503, −4.53434509519388340639219722607, −2.75062123749709962093078165071, −1.35795921922294321547072713129, 1.78752101784372838563881709381, 2.63904242292705613475051831101, 4.18041742797586052840275388863, 5.56262400445531744057112123902, 6.63310122311892209519594318311, 7.48210289346774768966317349248, 8.457382834439225665903971235690, 9.365695608946997222861982972484, 9.978771253867809022272812194734, 10.95074125898553509320008199060

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