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Properties

Label 2-560-20.3-c1-0-5
Degree $2$
Conductor $560$
Sign $-0.0368 - 0.999i$
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  + (−0.636 + 0.636i)3-s + (1.68 + 1.47i)5-s + (−0.707 − 0.707i)7-s + 2.19i·9-s − 0.0126i·11-s + (1.39 + 1.39i)13-s + (−2.00 + 0.134i)15-s + (−1.83 + 1.83i)17-s + 1.25·19-s + 0.899·21-s + (−0.761 + 0.761i)23-s + (0.666 + 4.95i)25-s + (−3.30 − 3.30i)27-s − 1.27i·29-s + 8.86i·31-s + ⋯
L(s)  = 1  + (−0.367 + 0.367i)3-s + (0.752 + 0.658i)5-s + (−0.267 − 0.267i)7-s + 0.730i·9-s − 0.00380i·11-s + (0.386 + 0.386i)13-s + (−0.518 + 0.0346i)15-s + (−0.444 + 0.444i)17-s + 0.288·19-s + 0.196·21-s + (−0.158 + 0.158i)23-s + (0.133 + 0.991i)25-s + (−0.635 − 0.635i)27-s − 0.237i·29-s + 1.59i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0368 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0368 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.0368 - 0.999i$
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: \((2,\ 560,\ (\ :1/2),\ -0.0368 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.902501 + 0.936391i\)
\(L(\frac12)\) \(\approx\) \(0.902501 + 0.936391i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 1.47i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (0.636 - 0.636i)T - 3iT^{2} \)
11 \( 1 + 0.0126iT - 11T^{2} \)
13 \( 1 + (-1.39 - 1.39i)T + 13iT^{2} \)
17 \( 1 + (1.83 - 1.83i)T - 17iT^{2} \)
19 \( 1 - 1.25T + 19T^{2} \)
23 \( 1 + (0.761 - 0.761i)T - 23iT^{2} \)
29 \( 1 + 1.27iT - 29T^{2} \)
31 \( 1 - 8.86iT - 31T^{2} \)
37 \( 1 + (0.649 - 0.649i)T - 37iT^{2} \)
41 \( 1 - 5.52T + 41T^{2} \)
43 \( 1 + (4.70 - 4.70i)T - 43iT^{2} \)
47 \( 1 + (-2.07 - 2.07i)T + 47iT^{2} \)
53 \( 1 + (2.69 + 2.69i)T + 53iT^{2} \)
59 \( 1 - 0.236T + 59T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 + (4.86 + 4.86i)T + 67iT^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 + (4.87 + 4.87i)T + 73iT^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 + (-12.2 + 12.2i)T - 83iT^{2} \)
89 \( 1 + 8.25iT - 89T^{2} \)
97 \( 1 + (-4.81 + 4.81i)T - 97iT^{2} \)
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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

−10.84688260495090118828613589984, −10.23896914399086583997849885134, −9.461881772634224199173458709177, −8.392058712992133865396431815776, −7.24147508040200217375309027084, −6.38728560797327909655232153063, −5.51724783526132760709828112830, −4.45167675264081188788593648218, −3.17169251530805870578072012883, −1.82901630497717202755301688976, 0.799892612061275608346684821512, 2.32125715743123041007343943469, 3.81077726180133843141616181067, 5.15544961276251213449735623122, 5.97380737156747401999994212616, 6.68706989188256976143645325911, 7.88847166092531967606851278577, 8.999880788315812040350461541303, 9.491891245510463309137931132829, 10.49487928469039157150226907813

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