↑

Properties

Label 2-360-8.5-c5-0-70
Degree $2$
Conductor $360$
Sign $0.999 + 0.0262i$
Analytic cond. $57.7381$
Root an. cond. $7.59856$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0494 + 5.65i)2-s + (−31.9 − 0.559i)4-s − 25i·5-s + 198.·7-s + (4.74 − 180. i)8-s + (141. + 1.23i)10-s − 85.9i·11-s + 407. i·13-s + (−9.83 + 1.12e3i)14-s + (1.02e3 + 35.8i)16-s − 1.20e3·17-s − 206. i·19-s + (−13.9 + 799. i)20-s + (486. + 4.25i)22-s + 2.59e3·23-s + ⋯
L(s)  = 1  + (−0.00874 + 0.999i)2-s + (−0.999 − 0.0174i)4-s − 0.447i·5-s + 1.53·7-s + (0.0262 − 0.999i)8-s + (0.447 + 0.00391i)10-s − 0.214i·11-s + 0.668i·13-s + (−0.0134 + 1.53i)14-s + (0.999 + 0.0349i)16-s − 1.01·17-s − 0.130i·19-s + (−0.00782 + 0.447i)20-s + (0.214 + 0.00187i)22-s + 1.02·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0262i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.999 + 0.0262i$
Analytic conductor: \(57.7381\)
Root analytic conductor: \(7.59856\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :5/2),\ 0.999 + 0.0262i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.924921766\)
\(L(\frac12)\) \(\approx\) \(1.924921766\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0494 - 5.65i)T \)
3 \( 1 \)
5 \( 1 + 25iT \)
good7 \( 1 - 198.T + 1.68e4T^{2} \)
11 \( 1 + 85.9iT - 1.61e5T^{2} \)
13 \( 1 - 407. iT - 3.71e5T^{2} \)
17 \( 1 + 1.20e3T + 1.41e6T^{2} \)
19 \( 1 + 206. iT - 2.47e6T^{2} \)
23 \( 1 - 2.59e3T + 6.43e6T^{2} \)
29 \( 1 + 6.19e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.86e3T + 2.86e7T^{2} \)
37 \( 1 + 1.47e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.80e4T + 1.15e8T^{2} \)
43 \( 1 + 9.26e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.43e4T + 2.29e8T^{2} \)
53 \( 1 - 1.27e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.07e4iT - 7.14e8T^{2} \)
61 \( 1 - 1.13e4iT - 8.44e8T^{2} \)
67 \( 1 + 6.26e4iT - 1.35e9T^{2} \)
71 \( 1 - 6.12e4T + 1.80e9T^{2} \)
73 \( 1 - 2.32e4T + 2.07e9T^{2} \)
79 \( 1 - 2.91e4T + 3.07e9T^{2} \)
83 \( 1 + 4.80e4iT - 3.93e9T^{2} \)
89 \( 1 + 3.01e4T + 5.58e9T^{2} \)
97 \( 1 + 1.13e5T + 8.58e9T^{2} \)
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

−10.63866771564895773068925409511, −9.207505912876866025922053598695, −8.701147280378866582427871061039, −7.78271732648192505379706916668, −6.92151346188581952015078206987, −5.67718730546226850424911224624, −4.80425265751403002378814033412, −4.04745019294080816311982562648, −1.95440336408180022414280716355, −0.56837232731001444311546250707, 1.10661037381552663639379905181, 2.14054114694982481272374533861, 3.32858517442270142746557632790, 4.64576951672619156391431620393, 5.29677913063343848198145111829, 6.94065158307818114935202555508, 8.127281224396205301755075977970, 8.751310875897942274834480695190, 9.957774355286846867425882626583, 10.90122402834025062874547547158

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