Properties

Label 2-580-580.399-c0-0-2
Degree $2$
Conductor $580$
Sign $0.857 + 0.514i$
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  + (−0.974 + 0.222i)2-s + (0.193 − 0.400i)3-s + (0.900 − 0.433i)4-s + (−0.222 − 0.974i)5-s + (−0.0990 + 0.433i)6-s + (0.781 + 0.376i)7-s + (−0.781 + 0.623i)8-s + (0.499 + 0.626i)9-s + (0.433 + 0.900i)10-s − 0.445i·12-s + (−0.846 − 0.193i)14-s + (−0.433 − 0.0990i)15-s + (0.623 − 0.781i)16-s + (−0.626 − 0.5i)18-s + (−0.623 − 0.781i)20-s + (0.301 − 0.240i)21-s + ⋯
L(s)  = 1  + (−0.974 + 0.222i)2-s + (0.193 − 0.400i)3-s + (0.900 − 0.433i)4-s + (−0.222 − 0.974i)5-s + (−0.0990 + 0.433i)6-s + (0.781 + 0.376i)7-s + (−0.781 + 0.623i)8-s + (0.499 + 0.626i)9-s + (0.433 + 0.900i)10-s − 0.445i·12-s + (−0.846 − 0.193i)14-s + (−0.433 − 0.0990i)15-s + (0.623 − 0.781i)16-s + (−0.626 − 0.5i)18-s + (−0.623 − 0.781i)20-s + (0.301 − 0.240i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(580\)    =    \(2^{2} \cdot 5 \cdot 29\)
Sign: $0.857 + 0.514i$
Analytic conductor: \(0.289457\)
Root analytic conductor: \(0.538012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{580} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 580,\ (\ :0),\ 0.857 + 0.514i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6889525362\)
\(L(\frac12)\) \(\approx\) \(0.6889525362\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.974 - 0.222i)T \)
5 \( 1 + (0.222 + 0.974i)T \)
29 \( 1 + (0.900 + 0.433i)T \)
good3 \( 1 + (-0.193 + 0.400i)T + (-0.623 - 0.781i)T^{2} \)
7 \( 1 + (-0.781 - 0.376i)T + (0.623 + 0.781i)T^{2} \)
11 \( 1 + (-0.222 - 0.974i)T^{2} \)
13 \( 1 + (0.222 + 0.974i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (0.623 - 0.781i)T^{2} \)
23 \( 1 + (-0.433 + 1.90i)T + (-0.900 - 0.433i)T^{2} \)
31 \( 1 + (-0.900 + 0.433i)T^{2} \)
37 \( 1 + (-0.222 + 0.974i)T^{2} \)
41 \( 1 - 1.56iT - T^{2} \)
43 \( 1 + (1.21 + 0.277i)T + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (-1.40 - 1.12i)T + (0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.900 - 0.433i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.623 - 0.781i)T^{2} \)
67 \( 1 + (-0.222 + 0.974i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.900 - 0.433i)T^{2} \)
79 \( 1 + (-0.222 + 0.974i)T^{2} \)
83 \( 1 + (1.40 - 0.678i)T + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.846 - 0.193i)T + (0.900 - 0.433i)T^{2} \)
97 \( 1 + (0.623 - 0.781i)T^{2} \)
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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

−10.79002735777318564790033641005, −9.826866357842381703464985376255, −8.846102232936446464542612088643, −8.237432893457490916446519171728, −7.64558160801192591735699598093, −6.57726734592523807486116634834, −5.37081875464796997618845313046, −4.47903720308052091374692659062, −2.45145430766544914384929883387, −1.34909750926378692741777822648, 1.68327825381533313832490027152, 3.20876723768813770870134505877, 3.99763776195702522780180603344, 5.65681016446873473539310051870, 7.03834956618020962801681320608, 7.34835908903189585477425286498, 8.449246217761831815386254226595, 9.403135588632853078167523929469, 10.10176739996314834739860804725, 10.90402758121743881130074448958

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