↑

Properties

Label 2-480-12.11-c3-0-22
Degree $2$
Conductor $480$
Sign $0.713 + 0.701i$
Analytic cond. $28.3209$
Root an. cond. $5.32174$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−5.19 + 0.0444i)3-s − 5i·5-s + 5.27i·7-s + (26.9 − 0.461i)9-s + 36.5·11-s − 81.4·13-s + (0.222 + 25.9i)15-s − 56.6i·17-s + 124. i·19-s + (−0.234 − 27.4i)21-s + 71.2·23-s − 25·25-s + (−140. + 3.60i)27-s + 87.4i·29-s + 55.1i·31-s + ⋯
L(s)  = 1  + (−0.999 + 0.00855i)3-s − 0.447i·5-s + 0.285i·7-s + (0.999 − 0.0171i)9-s + 1.00·11-s − 1.73·13-s + (0.00382 + 0.447i)15-s − 0.808i·17-s + 1.49i·19-s + (−0.00243 − 0.285i)21-s + 0.646·23-s − 0.200·25-s + (−0.999 + 0.0256i)27-s + 0.560i·29-s + 0.319i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.701i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.713 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(480\)    =    \(2^{5} \cdot 3 \cdot 5\)
Sign: $0.713 + 0.701i$
Analytic conductor: \(28.3209\)
Root analytic conductor: \(5.32174\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{480} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 480,\ (\ :3/2),\ 0.713 + 0.701i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.159542626\)
\(L(\frac12)\) \(\approx\) \(1.159542626\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (5.19 - 0.0444i)T \)
5 \( 1 + 5iT \)
good7 \( 1 - 5.27iT - 343T^{2} \)
11 \( 1 - 36.5T + 1.33e3T^{2} \)
13 \( 1 + 81.4T + 2.19e3T^{2} \)
17 \( 1 + 56.6iT - 4.91e3T^{2} \)
19 \( 1 - 124. iT - 6.85e3T^{2} \)
23 \( 1 - 71.2T + 1.21e4T^{2} \)
29 \( 1 - 87.4iT - 2.43e4T^{2} \)
31 \( 1 - 55.1iT - 2.97e4T^{2} \)
37 \( 1 - 7.97T + 5.06e4T^{2} \)
41 \( 1 + 247. iT - 6.89e4T^{2} \)
43 \( 1 + 407. iT - 7.95e4T^{2} \)
47 \( 1 - 415.T + 1.03e5T^{2} \)
53 \( 1 + 714. iT - 1.48e5T^{2} \)
59 \( 1 - 341.T + 2.05e5T^{2} \)
61 \( 1 + 191.T + 2.26e5T^{2} \)
67 \( 1 + 169. iT - 3.00e5T^{2} \)
71 \( 1 - 249.T + 3.57e5T^{2} \)
73 \( 1 - 1.15e3T + 3.89e5T^{2} \)
79 \( 1 + 501. iT - 4.93e5T^{2} \)
83 \( 1 + 239.T + 5.71e5T^{2} \)
89 \( 1 + 382. iT - 7.04e5T^{2} \)
97 \( 1 - 551.T + 9.12e5T^{2} \)
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

−10.41790435954847040254866960775, −9.688002378975718613798629382616, −8.852105726505003805088249951927, −7.47513466831152552655638693830, −6.80448502582021999143455750777, −5.58007039731632746024554886682, −4.94581031999728134537200114824, −3.79870375705211138353157041369, −1.99828155765561222043139097973, −0.57476633536717184872333422193, 0.888541203679793248401493352363, 2.51094630637709658926002984625, 4.12479583820687205865677484954, 4.93408606666781627652394811051, 6.14892488635906751845846776021, 6.93683113162543492925214329059, 7.62019695574643921250845571798, 9.179058781008220593885780765398, 9.889867853156771712103031027034, 10.81626490781218825442563458280

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