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Properties

Label 2-560-20.3-c1-0-2
Degree $2$
Conductor $560$
Sign $-0.0372 - 0.999i$
Analytic cond. $4.47162$
Root an. cond. $2.11462$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.95 + 1.95i)3-s + (0.432 − 2.19i)5-s + (−0.707 − 0.707i)7-s − 4.62i·9-s + 1.68i·11-s + (4.19 + 4.19i)13-s + (3.43 + 5.12i)15-s + (−1.32 + 1.32i)17-s + 6.35·19-s + 2.76·21-s + (−5.12 + 5.12i)23-s + (−4.62 − 1.89i)25-s + (3.17 + 3.17i)27-s + 5.35i·29-s − 0.528i·31-s + ⋯
L(s)  = 1  + (−1.12 + 1.12i)3-s + (0.193 − 0.981i)5-s + (−0.267 − 0.267i)7-s − 1.54i·9-s + 0.509i·11-s + (1.16 + 1.16i)13-s + (0.888 + 1.32i)15-s + (−0.322 + 0.322i)17-s + 1.45·19-s + 0.602·21-s + (−1.06 + 1.06i)23-s + (−0.925 − 0.379i)25-s + (0.611 + 0.611i)27-s + 0.994i·29-s − 0.0949i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0372 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0372 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.0372 - 0.999i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.0372 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.618106 + 0.641576i\)
\(L(\frac12)\) \(\approx\) \(0.618106 + 0.641576i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.432 + 2.19i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (1.95 - 1.95i)T - 3iT^{2} \)
11 \( 1 - 1.68iT - 11T^{2} \)
13 \( 1 + (-4.19 - 4.19i)T + 13iT^{2} \)
17 \( 1 + (1.32 - 1.32i)T - 17iT^{2} \)
19 \( 1 - 6.35T + 19T^{2} \)
23 \( 1 + (5.12 - 5.12i)T - 23iT^{2} \)
29 \( 1 - 5.35iT - 29T^{2} \)
31 \( 1 + 0.528iT - 31T^{2} \)
37 \( 1 + (5.76 - 5.76i)T - 37iT^{2} \)
41 \( 1 - 0.658T + 41T^{2} \)
43 \( 1 + (-2.91 + 2.91i)T - 43iT^{2} \)
47 \( 1 + (-8.83 - 8.83i)T + 47iT^{2} \)
53 \( 1 + (-7.38 - 7.38i)T + 53iT^{2} \)
59 \( 1 + 2.97T + 59T^{2} \)
61 \( 1 - 8.77T + 61T^{2} \)
67 \( 1 + (-4.37 - 4.37i)T + 67iT^{2} \)
71 \( 1 + 7.81iT - 71T^{2} \)
73 \( 1 + (0.896 + 0.896i)T + 73iT^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + (5.65 - 5.65i)T - 83iT^{2} \)
89 \( 1 - 6.27iT - 89T^{2} \)
97 \( 1 + (-10.3 + 10.3i)T - 97iT^{2} \)
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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

−10.99913597626413137373885670311, −10.04869944939902458153176169061, −9.438515063163713544999088770272, −8.690079641061128777628752526318, −7.26647042623569432628596713468, −6.07467347773974707599202368838, −5.41268580477808993613884521415, −4.41654109709810006784312595615, −3.74892334334488478988953408618, −1.36997401477313245607967325344, 0.65570882796859915835277667498, 2.33305747927781613378818666283, 3.60479282664182642215846970227, 5.55361001633701729285671071556, 5.91685318916549494596159072545, 6.80349898296987721857613956996, 7.58847324096733469804337227636, 8.557864042954544665632951759771, 10.00319247388420407890761046957, 10.72174313189197187504383556863

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