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Properties

Label 2-360-72.11-c1-0-46
Degree $2$
Conductor $360$
Sign $-0.985 - 0.167i$
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  + (0.884 − 1.10i)2-s + (−1.18 − 1.26i)3-s + (−0.436 − 1.95i)4-s + (0.5 + 0.866i)5-s + (−2.44 + 0.193i)6-s + (−2.20 − 1.27i)7-s + (−2.54 − 1.24i)8-s + (−0.185 + 2.99i)9-s + (1.39 + 0.213i)10-s + (−4.02 − 2.32i)11-s + (−1.94 + 2.86i)12-s + (3.23 − 1.87i)13-s + (−3.35 + 1.30i)14-s + (0.499 − 1.65i)15-s + (−3.61 + 1.70i)16-s − 3.38i·17-s + ⋯
L(s)  = 1  + (0.625 − 0.780i)2-s + (−0.684 − 0.728i)3-s + (−0.218 − 0.975i)4-s + (0.223 + 0.387i)5-s + (−0.996 + 0.0790i)6-s + (−0.834 − 0.481i)7-s + (−0.898 − 0.439i)8-s + (−0.0616 + 0.998i)9-s + (0.442 + 0.0676i)10-s + (−1.21 − 0.701i)11-s + (−0.561 + 0.827i)12-s + (0.898 − 0.518i)13-s + (−0.897 + 0.350i)14-s + (0.129 − 0.428i)15-s + (−0.904 + 0.425i)16-s − 0.821i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.985 - 0.167i$
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: \((2,\ 360,\ (\ :1/2),\ -0.985 - 0.167i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0819104 + 0.972042i\)
\(L(\frac12)\) \(\approx\) \(0.0819104 + 0.972042i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.884 + 1.10i)T \)
3 \( 1 + (1.18 + 1.26i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (2.20 + 1.27i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.02 + 2.32i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.23 + 1.87i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.38iT - 17T^{2} \)
19 \( 1 + 5.12T + 19T^{2} \)
23 \( 1 + (-4.16 - 7.21i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.01 + 3.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.88 + 2.81i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.40iT - 37T^{2} \)
41 \( 1 + (5.23 - 3.02i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.26 + 9.11i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.621 + 1.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.05T + 53T^{2} \)
59 \( 1 + (0.351 - 0.202i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.57 + 2.64i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.17 + 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.43T + 71T^{2} \)
73 \( 1 - 13.3T + 73T^{2} \)
79 \( 1 + (-5.40 - 3.11i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-12.0 - 6.97i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.72iT - 89T^{2} \)
97 \( 1 + (-1.31 + 2.27i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.77583286334407688364163473335, −10.67605043916397973207415979873, −9.440796757066990159118089150541, −8.016955348555688499577847887740, −6.79595875472793271825732076105, −5.97201303197484429576175633615, −5.16024422104248633471356744002, −3.55355037822099307445296895504, −2.43258799816357287057320704301, −0.57385558354251771192100431296, 2.89039164528176000762507190572, 4.29901101943148576032549579370, 5.03521994697749524248614899342, 6.19962654620009329415832868251, 6.63606612295618832184302566923, 8.329719070396318947860751052521, 8.980195502594808151509591521441, 10.14454588155614519248632458024, 10.96076579666978098904354406140, 12.37307225629358957282894944322

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