↑

Properties

Label 2-360-120.59-c1-0-12
Degree $2$
Conductor $360$
Sign $0.168 + 0.985i$
Analytic cond. $2.87461$
Root an. cond. $1.69546$
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  + (−0.927 − 1.06i)2-s + (−0.280 + 1.98i)4-s + (2.18 − 0.468i)5-s − 3.02·7-s + (2.37 − 1.53i)8-s + (−2.52 − 1.90i)10-s − 3.62i·11-s + 1.69·13-s + (2.80 + 3.22i)14-s + (−3.84 − 1.11i)16-s + 6.60·17-s + 5.12·19-s + (0.313 + 4.46i)20-s + (−3.86 + 3.35i)22-s − 6.67i·23-s + ⋯
L(s)  = 1  + (−0.655 − 0.755i)2-s + (−0.140 + 0.990i)4-s + (0.977 − 0.209i)5-s − 1.14·7-s + (0.839 − 0.543i)8-s + (−0.799 − 0.601i)10-s − 1.09i·11-s + 0.470·13-s + (0.748 + 0.862i)14-s + (−0.960 − 0.277i)16-s + 1.60·17-s + 1.17·19-s + (0.0700 + 0.997i)20-s + (−0.824 + 0.716i)22-s − 1.39i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.168 + 0.985i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.168 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.790739 - 0.666846i\)
\(L(\frac12)\) \(\approx\) \(0.790739 - 0.666846i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.927 + 1.06i)T \)
3 \( 1 \)
5 \( 1 + (-2.18 + 0.468i)T \)
good7 \( 1 + 3.02T + 7T^{2} \)
11 \( 1 + 3.62iT - 11T^{2} \)
13 \( 1 - 1.69T + 13T^{2} \)
17 \( 1 - 6.60T + 17T^{2} \)
19 \( 1 - 5.12T + 19T^{2} \)
23 \( 1 + 6.67iT - 23T^{2} \)
29 \( 1 + 6.82T + 29T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 - 0.371T + 37T^{2} \)
41 \( 1 + 5.83iT - 41T^{2} \)
43 \( 1 - 5.24iT - 43T^{2} \)
47 \( 1 + 0.525iT - 47T^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 + 4.86iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 13.4iT - 67T^{2} \)
71 \( 1 - 2.45T + 71T^{2} \)
73 \( 1 + 14.5iT - 73T^{2} \)
79 \( 1 - 14.1iT - 79T^{2} \)
83 \( 1 + 5.79T + 83T^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 - 9.33iT - 97T^{2} \)
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

−11.04273683203033943846084268998, −10.17574376607636312817070107573, −9.524721007603020388240603248725, −8.791182957631213000837274626854, −7.68355722680627798121996097768, −6.40458632240961003805441906202, −5.47499946514492266866361770922, −3.65018356307988291462622873205, −2.76485086908052726299913746365, −1.00226507219508147246909042636, 1.56752507599447489709505913975, 3.34584310041971173154390917027, 5.25615337326152516836115643851, 5.92330060708923938140813915392, 6.97891614503539260050690193226, 7.67605925998647901042035057119, 9.161175775933300561455052521386, 9.785961067933051401001471550212, 10.12713290737307570561180822029, 11.47713863662869034260935158051

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