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Properties

Label 2-360-72.11-c1-0-12
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: $\chi_{360} (11, \cdot )$
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

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

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