Properties

Label 2-560-7.2-c1-0-1
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.207 + 0.358i)3-s + (−0.5 + 0.866i)5-s + (−1.62 + 2.09i)7-s + (1.41 − 2.44i)9-s + (0.414 + 0.717i)11-s + 2·13-s − 0.414·15-s + (3.82 + 6.63i)17-s + (−2.82 + 4.89i)19-s + (−1.08 − 0.148i)21-s + (−2.79 + 4.83i)23-s + (−0.499 − 0.866i)25-s + 2.41·27-s − 7.82·29-s + (0.414 + 0.717i)31-s + ⋯
L(s)  = 1  + (0.119 + 0.207i)3-s + (−0.223 + 0.387i)5-s + (−0.612 + 0.790i)7-s + (0.471 − 0.816i)9-s + (0.124 + 0.216i)11-s + 0.554·13-s − 0.106·15-s + (0.928 + 1.60i)17-s + (−0.648 + 1.12i)19-s + (−0.236 − 0.0324i)21-s + (−0.582 + 1.00i)23-s + (−0.0999 − 0.173i)25-s + 0.464·27-s − 1.45·29-s + (0.0743 + 0.128i)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: $\chi_{560} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.0725 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.942972 + 0.876858i\)
\(L(\frac12)\) \(\approx\) \(0.942972 + 0.876858i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.207 - 0.358i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-0.414 - 0.717i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-3.82 - 6.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.82 - 4.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.79 - 4.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.82T + 29T^{2} \)
31 \( 1 + (-0.414 - 0.717i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.82 - 4.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.82T + 41T^{2} \)
43 \( 1 - 6.89T + 43T^{2} \)
47 \( 1 + (-5.82 + 10.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.82 - 4.89i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.32 - 5.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.44 + 11.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (1.82 + 3.16i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.75T + 83T^{2} \)
89 \( 1 + (-2.67 + 4.62i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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

−10.81530735163101158921221463325, −10.05794791965283607599830902375, −9.300045412721977328751124675250, −8.390327831863284434939750909412, −7.42671116477326197263980006442, −6.20514548278450681386118297719, −5.77296498686735187792720309104, −3.93604656073639836598130336332, −3.47148343844777682612497715294, −1.77476745804642224710503056426, 0.75983506415339821548267893998, 2.52688002110971297911429488291, 3.88671544927220471547075621835, 4.79871513576928201258031087100, 6.00678235773738969096745108412, 7.20226220740905371285949491999, 7.65429634246921221814901801122, 8.863398252761172061584058143452, 9.622047231357248625874364032738, 10.64792163583675864162293246175

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