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Properties

Label 2-360-72.59-c1-0-34
Degree $2$
Conductor $360$
Sign $0.922 - 0.384i$
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  + (1.38 − 0.300i)2-s + (1.28 + 1.16i)3-s + (1.81 − 0.829i)4-s + (−0.5 + 0.866i)5-s + (2.12 + 1.21i)6-s + (−1.13 + 0.653i)7-s + (2.26 − 1.69i)8-s + (0.306 + 2.98i)9-s + (−0.430 + 1.34i)10-s + (0.447 − 0.258i)11-s + (3.30 + 1.04i)12-s + (−1.21 − 0.701i)13-s + (−1.36 + 1.24i)14-s + (−1.64 + 0.533i)15-s + (2.62 − 3.02i)16-s − 2.63i·17-s + ⋯
L(s)  = 1  + (0.977 − 0.212i)2-s + (0.742 + 0.670i)3-s + (0.909 − 0.414i)4-s + (−0.223 + 0.387i)5-s + (0.867 + 0.497i)6-s + (−0.427 + 0.246i)7-s + (0.800 − 0.598i)8-s + (0.102 + 0.994i)9-s + (−0.136 + 0.425i)10-s + (0.134 − 0.0778i)11-s + (0.953 + 0.301i)12-s + (−0.337 − 0.194i)13-s + (−0.365 + 0.331i)14-s + (−0.425 + 0.137i)15-s + (0.655 − 0.755i)16-s − 0.638i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.922 - 0.384i$
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: \((2,\ 360,\ (\ :1/2),\ 0.922 - 0.384i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.71033 + 0.542633i\)
\(L(\frac12)\) \(\approx\) \(2.71033 + 0.542633i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.300i)T \)
3 \( 1 + (-1.28 - 1.16i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (1.13 - 0.653i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.447 + 0.258i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.21 + 0.701i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.63iT - 17T^{2} \)
19 \( 1 + 2.33T + 19T^{2} \)
23 \( 1 + (-2.91 + 5.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.53 - 2.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (8.36 + 4.82i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.68iT - 37T^{2} \)
41 \( 1 + (2.75 + 1.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.15 - 2.00i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.479 + 0.830i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.64T + 53T^{2} \)
59 \( 1 + (7.90 + 4.56i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.31 + 4.80i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.18 - 12.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.93T + 71T^{2} \)
73 \( 1 - 7.66T + 73T^{2} \)
79 \( 1 + (13.5 - 7.82i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.39 + 4.84i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.47iT - 89T^{2} \)
97 \( 1 + (2.47 + 4.29i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.44641365337939865941326639080, −10.66708817118692496581634665591, −9.854577355404236432676035507817, −8.838651039379139053596225640210, −7.58996326551549665478401804834, −6.64391813397020716820963376316, −5.36647302198327890651019980445, −4.33659705715710336392631554916, −3.28186922640770359910347903438, −2.37786861786718042676502321162, 1.80591910338135968037438376618, 3.25079548552314128466090766914, 4.15733434591274364652020105184, 5.54965358785467383390701479977, 6.70406485581077134042407514825, 7.39325782814398291615274240934, 8.369830043853807835348037666122, 9.360661909912408467324804833536, 10.65669205173050820362622725597, 11.79781494774274000971288557259

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