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Properties

Label 2-560-7.4-c1-0-13
Degree $2$
Conductor $560$
Sign $0.0725 + 0.997i$
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  + (1.20 − 2.09i)3-s + (−0.5 − 0.866i)5-s + (1.62 + 2.09i)7-s + (−1.41 − 2.44i)9-s + (2.41 − 4.18i)11-s + 0.828·13-s − 2.41·15-s + (0.414 − 0.717i)17-s + (−1.41 − 2.44i)19-s + (6.32 − 0.866i)21-s + (−1.20 − 2.09i)23-s + (−0.499 + 0.866i)25-s + 0.414·27-s − 29-s + (−3 + 5.19i)31-s + ⋯
L(s)  = 1  + (0.696 − 1.20i)3-s + (−0.223 − 0.387i)5-s + (0.612 + 0.790i)7-s + (−0.471 − 0.816i)9-s + (0.727 − 1.26i)11-s + 0.229·13-s − 0.623·15-s + (0.100 − 0.174i)17-s + (−0.324 − 0.561i)19-s + (1.38 − 0.188i)21-s + (−0.251 − 0.435i)23-s + (−0.0999 + 0.173i)25-s + 0.0797·27-s − 0.185·29-s + (−0.538 + 0.933i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.0725 + 0.997i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.0725 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38399 - 1.28696i\)
\(L(\frac12)\) \(\approx\) \(1.38399 - 1.28696i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.62 - 2.09i)T \)
good3 \( 1 + (-1.20 + 2.09i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2.41 + 4.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.828T + 13T^{2} \)
17 \( 1 + (-0.414 + 0.717i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.41 + 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.20 + 2.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.17T + 41T^{2} \)
43 \( 1 + 6.41T + 43T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.41 + 5.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.24 - 10.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.74 - 9.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.20 + 10.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + (2.41 - 4.18i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.58 - 7.94i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + (1.32 + 2.30i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.343T + 97T^{2} \)
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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

−10.77749571473958819842230476997, −9.215156087650367162890581279072, −8.566481787800159991827043593382, −8.137100412961582865089278851545, −7.01521854175298330302771318043, −6.14117670405971565415073048067, −5.03551250295328867166217682420, −3.54887764519543693343909658521, −2.34469673081357420926078807609, −1.14074371555378411590540201457, 1.91784945743143448103111858765, 3.59298966061989341873598265011, 4.10195165077672420194529862773, 5.03743502284710335031860755556, 6.56463653718944835422696793039, 7.55915689093265477285642757205, 8.366024452716653283551116191212, 9.461293011947696529416382109469, 9.989832758248333588901473778051, 10.77183777489182139260513220839

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