↑

Properties

Label 2-280-35.9-c1-0-1
Degree $2$
Conductor $280$
Sign $-0.0605 - 0.998i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
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  + (−0.650 + 0.375i)3-s + (−0.702 + 2.12i)5-s + (0.543 − 2.58i)7-s + (−1.21 + 2.10i)9-s + (2.63 + 4.57i)11-s + 2.43i·13-s + (−0.340 − 1.64i)15-s + (−5.30 + 3.06i)17-s + (−0.220 + 0.382i)19-s + (0.619 + 1.88i)21-s + (2.75 + 1.59i)23-s + (−4.01 − 2.98i)25-s − 4.08i·27-s + 1.25·29-s + (0.322 + 0.558i)31-s + ⋯
L(s)  = 1  + (−0.375 + 0.216i)3-s + (−0.314 + 0.949i)5-s + (0.205 − 0.978i)7-s + (−0.405 + 0.703i)9-s + (0.795 + 1.37i)11-s + 0.675i·13-s + (−0.0878 − 0.424i)15-s + (−1.28 + 0.742i)17-s + (−0.0506 + 0.0876i)19-s + (0.135 + 0.412i)21-s + (0.575 + 0.332i)23-s + (−0.802 − 0.596i)25-s − 0.786i·27-s + 0.232·29-s + (0.0579 + 0.100i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0605 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0605 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.0605 - 0.998i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.0605 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.657823 + 0.698902i\)
\(L(\frac12)\) \(\approx\) \(0.657823 + 0.698902i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.702 - 2.12i)T \)
7 \( 1 + (-0.543 + 2.58i)T \)
good3 \( 1 + (0.650 - 0.375i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2.63 - 4.57i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.43iT - 13T^{2} \)
17 \( 1 + (5.30 - 3.06i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.220 - 0.382i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.75 - 1.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.25T + 29T^{2} \)
31 \( 1 + (-0.322 - 0.558i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.31 - 5.37i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.90T + 41T^{2} \)
43 \( 1 + 10.4iT - 43T^{2} \)
47 \( 1 + (6.52 + 3.76i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.89 + 1.67i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.07 + 7.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.73 + 3.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.14 - 2.39i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 + (4.61 - 2.66i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.70 + 4.68i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.9iT - 83T^{2} \)
89 \( 1 + (-4.30 + 7.44i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.6iT - 97T^{2} \)
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

−11.78008656901452722441613395235, −11.11341729630333129585661404762, −10.43324768008073265012173472515, −9.474641344721556414462045496284, −8.079742498001572844914973569433, −7.06897078647197141316368344007, −6.42614664467515254003538923189, −4.70378343202025219258815121383, −3.94530337289070819371856523440, −2.12071680655395158995471239965, 0.78688607544034668743157994144, 2.92781827401876163410759409434, 4.46892845462423681547386014596, 5.68461242340301537110856278180, 6.36108522344683497077560946887, 7.936005003171993165654737464649, 8.950366232611689201556555514123, 9.232498034840121359995982361064, 11.24730272380920869021916277029, 11.44404319302429583407504489345

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