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Properties

Label 2-560-80.29-c1-0-51
Degree $2$
Conductor $560$
Sign $-0.655 + 0.755i$
Analytic cond. $4.47162$
Root an. cond. $2.11462$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (−1 + i)3-s − 2i·4-s + (−1 − 2i)5-s − 2i·6-s + 7-s + (2 + 2i)8-s + i·9-s + (3 + i)10-s + (1 − i)11-s + (2 + 2i)12-s + (−3 + 3i)13-s + (−1 + i)14-s + (3 + i)15-s − 4·16-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.577 + 0.577i)3-s − i·4-s + (−0.447 − 0.894i)5-s − 0.816i·6-s + 0.377·7-s + (0.707 + 0.707i)8-s + 0.333i·9-s + (0.948 + 0.316i)10-s + (0.301 − 0.301i)11-s + (0.577 + 0.577i)12-s + (−0.832 + 0.832i)13-s + (−0.267 + 0.267i)14-s + (0.774 + 0.258i)15-s − 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.655 + 0.755i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.655 + 0.755i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
5 \( 1 + (1 + 2i)T \)
7 \( 1 - T \)
good3 \( 1 + (1 - i)T - 3iT^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
13 \( 1 + (3 - 3i)T - 13iT^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + (3 + 3i)T + 19iT^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (1 + i)T + 29iT^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (5 + 5i)T + 37iT^{2} \)
41 \( 1 - 12iT - 41T^{2} \)
43 \( 1 + (1 + i)T + 43iT^{2} \)
47 \( 1 + 10iT - 47T^{2} \)
53 \( 1 + (-3 - 3i)T + 53iT^{2} \)
59 \( 1 + (1 - i)T - 59iT^{2} \)
61 \( 1 + (7 + 7i)T + 61iT^{2} \)
67 \( 1 + (3 - 3i)T - 67iT^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (5 - 5i)T - 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−10.36198207910143868452656178943, −9.367388518723168305184030040380, −8.773006473738189680563539278802, −7.82221982634645228963644952572, −6.96559094590727659148030102404, −5.71843200308178030147583344025, −4.92173408827166798320902009903, −4.24593149366840319265672024091, −1.84831417776244272968014517288, 0, 1.75151869700390277834938640222, 3.09822321493158430231107950252, 4.16861259605725274387258399077, 5.73326828500251985233243200145, 6.97608401717528197695588646189, 7.40981490569013751150811715410, 8.398067639627249830401592268085, 9.491997576291117631391731084043, 10.46577793246840576941950435195

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