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Properties

Label 2-360-72.11-c1-0-2
Degree $2$
Conductor $360$
Sign $0.351 - 0.936i$
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  + (−1.14 − 0.826i)2-s + (−0.952 − 1.44i)3-s + (0.633 + 1.89i)4-s + (0.5 + 0.866i)5-s + (−0.103 + 2.44i)6-s + (−2.56 − 1.47i)7-s + (0.841 − 2.70i)8-s + (−1.18 + 2.75i)9-s + (0.142 − 1.40i)10-s + (2.28 + 1.31i)11-s + (2.14 − 2.72i)12-s + (−5.81 + 3.35i)13-s + (1.71 + 3.81i)14-s + (0.776 − 1.54i)15-s + (−3.19 + 2.40i)16-s + 1.13i·17-s + ⋯
L(s)  = 1  + (−0.811 − 0.584i)2-s + (−0.549 − 0.835i)3-s + (0.316 + 0.948i)4-s + (0.223 + 0.387i)5-s + (−0.0422 + 0.999i)6-s + (−0.968 − 0.559i)7-s + (0.297 − 0.954i)8-s + (−0.395 + 0.918i)9-s + (0.0449 − 0.444i)10-s + (0.688 + 0.397i)11-s + (0.618 − 0.785i)12-s + (−1.61 + 0.930i)13-s + (0.459 + 1.02i)14-s + (0.200 − 0.399i)15-s + (−0.799 + 0.600i)16-s + 0.274i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.351 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.351 - 0.936i$
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: \((2,\ 360,\ (\ :1/2),\ 0.351 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.256862 + 0.177830i\)
\(L(\frac12)\) \(\approx\) \(0.256862 + 0.177830i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.14 + 0.826i)T \)
3 \( 1 + (0.952 + 1.44i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (2.56 + 1.47i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.28 - 1.31i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.81 - 3.35i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.13iT - 17T^{2} \)
19 \( 1 + 1.90T + 19T^{2} \)
23 \( 1 + (-1.73 - 3.00i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.38 - 4.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.93 + 1.11i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.4iT - 37T^{2} \)
41 \( 1 + (-8.25 + 4.76i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.52 - 4.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.06 + 1.85i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.02T + 53T^{2} \)
59 \( 1 + (11.8 - 6.86i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.75 + 3.90i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.22 + 3.85i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 - 4.30T + 73T^{2} \)
79 \( 1 + (10.4 + 6.06i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.80 - 1.04i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.3iT - 89T^{2} \)
97 \( 1 + (4.06 - 7.04i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.66761610865704578685993984167, −10.66101452864396290347169569200, −9.845303258328322449463688405029, −9.107981198716540858434560706214, −7.63257883156123721150839459364, −6.99033869286929137833050567414, −6.34051002748946681340003255412, −4.50211762373160050562363703084, −2.95588692692005619412528308202, −1.65735665323154430560329881540, 0.28703110010737762935828457336, 2.75681988905915535783582497152, 4.56346826823470985005886134238, 5.65096584662718780959282810258, 6.27466287202633736455505935438, 7.47888724037376447092005149872, 8.835165291405154469781142071126, 9.420371628311928943825409114160, 10.06052812070158059182656493550, 10.96550381370461147397699613150

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