↑

Properties

Label 2-560-140.39-c2-0-0
Degree $2$
Conductor $560$
Sign $0.494 + 0.869i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.56 + 4.44i)3-s + (−1.78 + 4.67i)5-s + (−0.710 + 6.96i)7-s + (−8.69 − 15.0i)9-s + (−14.8 − 8.58i)11-s − 13.7i·13-s + (−16.1 − 19.9i)15-s + (2.70 + 1.56i)17-s + (−8.15 + 4.70i)19-s + (−29.1 − 21.0i)21-s + (21.1 + 36.6i)23-s + (−18.6 − 16.6i)25-s + 43.0·27-s + 8.83·29-s + (33.6 + 19.4i)31-s + ⋯
L(s)  = 1  + (−0.856 + 1.48i)3-s + (−0.357 + 0.934i)5-s + (−0.101 + 0.994i)7-s + (−0.965 − 1.67i)9-s + (−1.35 − 0.780i)11-s − 1.05i·13-s + (−1.07 − 1.32i)15-s + (0.159 + 0.0918i)17-s + (−0.429 + 0.247i)19-s + (−1.38 − 1.00i)21-s + (0.919 + 1.59i)23-s + (−0.745 − 0.666i)25-s + 1.59·27-s + 0.304·29-s + (1.08 + 0.627i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.494 + 0.869i$
Analytic conductor: \(15.2588\)
Root analytic conductor: \(3.90626\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1),\ 0.494 + 0.869i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.04340873967\)
\(L(\frac12)\) \(\approx\) \(0.04340873967\)
\(L(2)\) 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 \( 1 + (1.78 - 4.67i)T \)
7 \( 1 + (0.710 - 6.96i)T \)
good3 \( 1 + (2.56 - 4.44i)T + (-4.5 - 7.79i)T^{2} \)
11 \( 1 + (14.8 + 8.58i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 13.7iT - 169T^{2} \)
17 \( 1 + (-2.70 - 1.56i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (8.15 - 4.70i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-21.1 - 36.6i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 8.83T + 841T^{2} \)
31 \( 1 + (-33.6 - 19.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (32.3 - 18.6i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 27.9T + 1.68e3T^{2} \)
43 \( 1 + 40.5T + 1.84e3T^{2} \)
47 \( 1 + (28.6 + 49.6i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-4.93 - 2.84i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-60.7 - 35.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (34.2 + 59.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-4.23 + 7.33i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 121. iT - 5.04e3T^{2} \)
73 \( 1 + (-62.7 - 36.2i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (79.6 - 45.9i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 47.8T + 6.88e3T^{2} \)
89 \( 1 + (-44.0 - 76.2i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 82.1iT - 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.16477152921004014327088414701, −10.35765529262744834691977886476, −9.978725092274818930708023979344, −8.732653317124008057226919235753, −7.88577895156151405320651178364, −6.44654438722645406037398619300, −5.51954784097988231837416787286, −5.06231114094939027693101349956, −3.48756137310047085635248216596, −2.91005688154586964348608670864, 0.02162058305700826970370705551, 1.07667038004859949301702561598, 2.36173584158545694453881872993, 4.41801691242083453459418265138, 5.07711073217601457886809574608, 6.38517151455562868784824134779, 7.10873511941422272020715409843, 7.80220705800125656743164854359, 8.639326170589359706245056373193, 10.01006463412244115834505524368

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