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Properties

Label 2-560-140.83-c2-0-39
Degree $2$
Conductor $560$
Sign $0.796 + 0.604i$
Analytic cond. $15.2588$
Root an. cond. $3.90626$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.491 + 0.491i)3-s + (4.95 + 0.646i)5-s + (6.17 + 3.29i)7-s − 8.51i·9-s − 15.9i·11-s + (9.60 − 9.60i)13-s + (2.11 + 2.75i)15-s + (−21.9 − 21.9i)17-s + 2.96i·19-s + (1.41 + 4.65i)21-s + (−22.9 + 22.9i)23-s + (24.1 + 6.41i)25-s + (8.60 − 8.60i)27-s + 2.31i·29-s + 49.4·31-s + ⋯
L(s)  = 1  + (0.163 + 0.163i)3-s + (0.991 + 0.129i)5-s + (0.882 + 0.470i)7-s − 0.946i·9-s − 1.45i·11-s + (0.739 − 0.739i)13-s + (0.141 + 0.183i)15-s + (−1.29 − 1.29i)17-s + 0.155i·19-s + (0.0674 + 0.221i)21-s + (−0.997 + 0.997i)23-s + (0.966 + 0.256i)25-s + (0.318 − 0.318i)27-s + 0.0798i·29-s + 1.59·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.796 + 0.604i$
Analytic conductor: \(15.2588\)
Root analytic conductor: \(3.90626\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1),\ 0.796 + 0.604i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.418604562\)
\(L(\frac12)\) \(\approx\) \(2.418604562\)
\(L(2)\) 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 \( 1 + (-4.95 - 0.646i)T \)
7 \( 1 + (-6.17 - 3.29i)T \)
good3 \( 1 + (-0.491 - 0.491i)T + 9iT^{2} \)
11 \( 1 + 15.9iT - 121T^{2} \)
13 \( 1 + (-9.60 + 9.60i)T - 169iT^{2} \)
17 \( 1 + (21.9 + 21.9i)T + 289iT^{2} \)
19 \( 1 - 2.96iT - 361T^{2} \)
23 \( 1 + (22.9 - 22.9i)T - 529iT^{2} \)
29 \( 1 - 2.31iT - 841T^{2} \)
31 \( 1 - 49.4T + 961T^{2} \)
37 \( 1 + (9.99 - 9.99i)T - 1.36e3iT^{2} \)
41 \( 1 + 58.8iT - 1.68e3T^{2} \)
43 \( 1 + (29.2 - 29.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (19.1 - 19.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (-22.0 - 22.0i)T + 2.80e3iT^{2} \)
59 \( 1 - 19.1iT - 3.48e3T^{2} \)
61 \( 1 + 88.8iT - 3.72e3T^{2} \)
67 \( 1 + (-20.9 - 20.9i)T + 4.48e3iT^{2} \)
71 \( 1 - 80.0iT - 5.04e3T^{2} \)
73 \( 1 + (-28.2 + 28.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 57.9T + 6.24e3T^{2} \)
83 \( 1 + (-70.9 - 70.9i)T + 6.88e3iT^{2} \)
89 \( 1 + 44.5T + 7.92e3T^{2} \)
97 \( 1 + (-72.3 - 72.3i)T + 9.40e3iT^{2} \)
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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

−10.48884508568291031138125919870, −9.449604769028856209507597242241, −8.788251390881316702328561563249, −8.054026562001868476450080614567, −6.57424313572562953758657055096, −5.89859140037757363879132227754, −5.01090603421011498706028637896, −3.55234205883986526269851799886, −2.48394511535516566762217784478, −0.972400901754998280726070327524, 1.69746237375180560310658580890, 2.18199132655029799372623460193, 4.32199303339871095985429894763, 4.80009602282748330131099757432, 6.21374734388524389070094272520, 6.95189819311753873758823974988, 8.162595696423663108377536508195, 8.704053959006290482687903040246, 9.994200032700653619272685680068, 10.48343508071549263975422043452

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