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Properties

Label 2-360-72.59-c1-0-25
Degree $2$
Conductor $360$
Sign $0.481 + 0.876i$
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  + (0.0747 − 1.41i)2-s + (−0.478 + 1.66i)3-s + (−1.98 − 0.211i)4-s + (0.5 − 0.866i)5-s + (2.31 + 0.800i)6-s + (1.88 − 1.09i)7-s + (−0.446 + 2.79i)8-s + (−2.54 − 1.59i)9-s + (−1.18 − 0.770i)10-s + (2.33 − 1.34i)11-s + (1.30 − 3.20i)12-s + (1.45 + 0.842i)13-s + (−1.39 − 2.74i)14-s + (1.20 + 1.24i)15-s + (3.91 + 0.840i)16-s − 1.26i·17-s + ⋯
L(s)  = 1  + (0.0528 − 0.998i)2-s + (−0.276 + 0.961i)3-s + (−0.994 − 0.105i)4-s + (0.223 − 0.387i)5-s + (0.945 + 0.326i)6-s + (0.713 − 0.412i)7-s + (−0.158 + 0.987i)8-s + (−0.847 − 0.531i)9-s + (−0.374 − 0.243i)10-s + (0.703 − 0.406i)11-s + (0.376 − 0.926i)12-s + (0.404 + 0.233i)13-s + (−0.373 − 0.734i)14-s + (0.310 + 0.321i)15-s + (0.977 + 0.210i)16-s − 0.306i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.481 + 0.876i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.481 + 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11055 - 0.656608i\)
\(L(\frac12)\) \(\approx\) \(1.11055 - 0.656608i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0747 + 1.41i)T \)
3 \( 1 + (0.478 - 1.66i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (-1.88 + 1.09i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.33 + 1.34i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.45 - 0.842i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.26iT - 17T^{2} \)
19 \( 1 - 7.49T + 19T^{2} \)
23 \( 1 + (-3.13 + 5.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.81 + 3.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.27 + 3.62i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.85iT - 37T^{2} \)
41 \( 1 + (3.69 + 2.13i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.78 - 10.0i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.33 - 5.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.537T + 53T^{2} \)
59 \( 1 + (4.48 + 2.58i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.13 - 5.27i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.42 + 7.67i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.23T + 71T^{2} \)
73 \( 1 - 1.89T + 73T^{2} \)
79 \( 1 + (9.75 - 5.63i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.764 - 0.441i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.4iT - 89T^{2} \)
97 \( 1 + (4.16 + 7.20i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.34341582136364195564213369724, −10.51220483892155912571861570184, −9.473714044147443425818630538197, −9.008643711533119917927842125946, −7.86204051385800433218609104033, −6.04813381492297131614647532706, −4.98582630121037372183358609978, −4.22599831668362983212248602429, −3.07821534719300733831424890926, −1.13553129558716632015770957025, 1.50933315417697787143872308113, 3.48346633284870216275292347902, 5.25029678208652151128577074783, 5.75508316836002796767780683573, 7.08050938471346155885652573065, 7.44152280021345892921637731192, 8.634837866367608527534033877483, 9.394196532094412261613884039463, 10.80640126476256596314606681500, 11.77263970819397664863279818637

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