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Properties

Label 2-560-16.13-c1-0-17
Degree $2$
Conductor $560$
Sign $-0.875 - 0.483i$
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  + (0.504 + 1.32i)2-s + (2.22 + 2.22i)3-s + (−1.49 + 1.33i)4-s + (0.707 − 0.707i)5-s + (−1.81 + 4.05i)6-s − i·7-s + (−2.51 − 1.29i)8-s + 6.85i·9-s + (1.29 + 0.577i)10-s + (−1.33 + 1.33i)11-s + (−6.26 − 0.349i)12-s + (0.657 + 0.657i)13-s + (1.32 − 0.504i)14-s + 3.13·15-s + (0.444 − 3.97i)16-s − 1.24·17-s + ⋯
L(s)  = 1  + (0.356 + 0.934i)2-s + (1.28 + 1.28i)3-s + (−0.745 + 0.666i)4-s + (0.316 − 0.316i)5-s + (−0.740 + 1.65i)6-s − 0.377i·7-s + (−0.888 − 0.458i)8-s + 2.28i·9-s + (0.408 + 0.182i)10-s + (−0.401 + 0.401i)11-s + (−1.80 − 0.100i)12-s + (0.182 + 0.182i)13-s + (0.353 − 0.134i)14-s + 0.810·15-s + (0.111 − 0.993i)16-s − 0.302·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 - 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.875 - 0.483i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.875 - 0.483i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.599573 + 2.32825i\)
\(L(\frac12)\) \(\approx\) \(0.599573 + 2.32825i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.504 - 1.32i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + iT \)
good3 \( 1 + (-2.22 - 2.22i)T + 3iT^{2} \)
11 \( 1 + (1.33 - 1.33i)T - 11iT^{2} \)
13 \( 1 + (-0.657 - 0.657i)T + 13iT^{2} \)
17 \( 1 + 1.24T + 17T^{2} \)
19 \( 1 + (1.56 + 1.56i)T + 19iT^{2} \)
23 \( 1 - 3.20iT - 23T^{2} \)
29 \( 1 + (-3.83 - 3.83i)T + 29iT^{2} \)
31 \( 1 - 11.1T + 31T^{2} \)
37 \( 1 + (-4.06 + 4.06i)T - 37iT^{2} \)
41 \( 1 + 12.5iT - 41T^{2} \)
43 \( 1 + (2.18 - 2.18i)T - 43iT^{2} \)
47 \( 1 + 1.62T + 47T^{2} \)
53 \( 1 + (-7.72 + 7.72i)T - 53iT^{2} \)
59 \( 1 + (0.930 - 0.930i)T - 59iT^{2} \)
61 \( 1 + (6.14 + 6.14i)T + 61iT^{2} \)
67 \( 1 + (-8.05 - 8.05i)T + 67iT^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 + 11.1iT - 73T^{2} \)
79 \( 1 + 9.90T + 79T^{2} \)
83 \( 1 + (-4.56 - 4.56i)T + 83iT^{2} \)
89 \( 1 - 14.3iT - 89T^{2} \)
97 \( 1 + 2.65T + 97T^{2} \)
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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

−10.75798019108121678365172665267, −9.927500881963547044588273675567, −9.222589278935870737896257354778, −8.504399519093641701460098894764, −7.79821347480436788369835474558, −6.68629258611163958078129495327, −5.26813568386923999590034689247, −4.54396136868306742692265780751, −3.72871795028270900571359011319, −2.56243538723442894171561735638, 1.19454711193223037495632979402, 2.54469580372980406677274724027, 2.92664246744911173407268758774, 4.36840681215624747293820416489, 5.97765058050954290736858870534, 6.64599272059076993842019114526, 8.163450732827378526600706651726, 8.438682864005210861752652816176, 9.553469853419027029444838782631, 10.32042627676324430516783841988

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