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Properties

Label 2-360-72.59-c1-0-38
Degree $2$
Conductor $360$
Sign $0.277 + 0.960i$
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  + (−1.31 + 0.518i)2-s + (0.424 − 1.67i)3-s + (1.46 − 1.36i)4-s + (0.5 − 0.866i)5-s + (0.311 + 2.42i)6-s + (3.88 − 2.24i)7-s + (−1.21 + 2.55i)8-s + (−2.63 − 1.42i)9-s + (−0.208 + 1.39i)10-s + (1.05 − 0.606i)11-s + (−1.66 − 3.03i)12-s + (1.71 + 0.987i)13-s + (−3.94 + 4.96i)14-s + (−1.24 − 1.20i)15-s + (0.277 − 3.99i)16-s + 5.28i·17-s + ⋯
L(s)  = 1  + (−0.930 + 0.366i)2-s + (0.245 − 0.969i)3-s + (0.731 − 0.682i)4-s + (0.223 − 0.387i)5-s + (0.127 + 0.991i)6-s + (1.46 − 0.847i)7-s + (−0.430 + 0.902i)8-s + (−0.879 − 0.475i)9-s + (−0.0660 + 0.442i)10-s + (0.316 − 0.182i)11-s + (−0.482 − 0.876i)12-s + (0.474 + 0.273i)13-s + (−1.05 + 1.32i)14-s + (−0.320 − 0.311i)15-s + (0.0694 − 0.997i)16-s + 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.277 + 0.960i$
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: \((2,\ 360,\ (\ :1/2),\ 0.277 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.893365 - 0.671578i\)
\(L(\frac12)\) \(\approx\) \(0.893365 - 0.671578i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 - 0.518i)T \)
3 \( 1 + (-0.424 + 1.67i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (-3.88 + 2.24i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.05 + 0.606i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.71 - 0.987i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.28iT - 17T^{2} \)
19 \( 1 + 4.86T + 19T^{2} \)
23 \( 1 + (-1.40 + 2.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.20 + 7.28i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.04 - 3.49i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.71iT - 37T^{2} \)
41 \( 1 + (5.30 + 3.06i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.17 - 2.04i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.97 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.739T + 53T^{2} \)
59 \( 1 + (-1.70 - 0.986i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.81 - 2.77i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.58 + 2.74i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.40T + 71T^{2} \)
73 \( 1 + 6.69T + 73T^{2} \)
79 \( 1 + (11.2 - 6.49i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.00 - 2.89i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.71iT - 89T^{2} \)
97 \( 1 + (-8.61 - 14.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.09910654954566176070766645235, −10.44763085582450831039009803515, −8.987883756727089385046397406412, −8.329470550474897163542364552282, −7.74906366762380260080959638523, −6.65151869457183140107474226688, −5.80677298819836623539706077530, −4.27511078977925440618537039898, −2.07147903131750682298923262730, −1.12170193071329263348983341175, 1.95463388761218005444419556769, 3.11782653232544794653478932161, 4.58252844190642130171570036830, 5.72997878819973319953265898821, 7.18312445576856346281795031590, 8.364448312603128078043607697889, 8.820048580767687392032874364124, 9.781181247666045370795156384352, 10.69492350984870636398975238778, 11.38951663049400169951884304755

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