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Properties

Label 2-360-120.59-c1-0-13
Degree $2$
Conductor $360$
Sign $0.249 - 0.968i$
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

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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 + (1.52 + 2.76i)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 + (−1.54 + 4.19i)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 + (0.482 + 0.875i)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 + (−0.344 + 0.938i)20-s + (0.824 − 0.716i)22-s + 1.39i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.249 - 0.968i$
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: \((2,\ 360,\ (\ :1/2),\ 0.249 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73670 + 1.34575i\)
\(L(\frac12)\) \(\approx\) \(1.73670 + 1.34575i\)
\(L(\frac{3}{2})\) 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 \( 1 + 9.33iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.46876624449463907848715103510, −11.11584101070880319080755532665, −9.549116718444608133342572657960, −8.738411610346438107959119657053, −7.71648949845135433309287528638, −6.78517604147374584039744828199, −5.61364050124735744638003224449, −5.08301756843702046475124345359, −3.61607289766875547149178244059, −2.13624341081602281442419021355, 1.62515988277951906713058806243, 2.54894514897133194855472053044, 4.50486785359579443612138864569, 4.95212896810919405113163276411, 6.16310825960318586195727224147, 7.30711110780857846551093464107, 8.810712012441947776661365029202, 9.582949529990553051025455515481, 10.47513516405170671963685981375, 11.28018768877054263606671756809

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