↑

Properties

Label 2-360-120.59-c1-0-17
Degree $2$
Conductor $360$
Sign $-0.577 + 0.816i$
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  − 1.41i·2-s − 2.00·4-s − 2.23i·5-s + 5.16·7-s + 2.82i·8-s − 3.16·10-s − 3.05i·11-s − 0.837·13-s − 7.30i·14-s + 4.00·16-s − 6.32·19-s + 4.47i·20-s − 4.32·22-s − 4.47i·23-s − 5.00·25-s + 1.18i·26-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s − 0.999i·5-s + 1.95·7-s + 1.00i·8-s − 1.00·10-s − 0.921i·11-s − 0.232·13-s − 1.95i·14-s + 1.00·16-s − 1.45·19-s + 1.00i·20-s − 0.921·22-s − 0.932i·23-s − 1.00·25-s + 0.232i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.577 + 0.816i$
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: \((2,\ 360,\ (\ :1/2),\ -0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.629869 - 1.21681i\)
\(L(\frac12)\) \(\approx\) \(0.629869 - 1.21681i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
5 \( 1 + 2.23iT \)
good7 \( 1 - 5.16T + 7T^{2} \)
11 \( 1 + 3.05iT - 11T^{2} \)
13 \( 1 + 0.837T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 6.32T + 19T^{2} \)
23 \( 1 + 4.47iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 11.1T + 37T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 2.82iT - 47T^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 - 5.42iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 18.8iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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

−11.21130546800064766715699045534, −10.49717374000316387344986186844, −9.193179363151243867063843891720, −8.357063296207359870647422677333, −8.000546133294500011024937977168, −5.87280731580810929046588201734, −4.77327239885164791882427187029, −4.24796819523525105243727385726, −2.33096459146852303413944768077, −1.07141084064220810782728484833, 2.03426706669829684757354654562, 4.06632291382269473254388189261, 4.91827071955962623303016851457, 6.04860661235902808640955141362, 7.23678274452262029170038834542, 7.75832216393884114544831445259, 8.694831155950842723075545465211, 9.887074337190325004860436534525, 10.80840600783233602955675998070, 11.63758233056339426426740561351

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