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Properties

Label 2-360-72.59-c1-0-16
Degree $2$
Conductor $360$
Sign $-0.691 - 0.722i$
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  + (0.767 + 1.18i)2-s + (0.801 + 1.53i)3-s + (−0.822 + 1.82i)4-s + (−0.5 + 0.866i)5-s + (−1.20 + 2.13i)6-s + (4.07 − 2.35i)7-s + (−2.79 + 0.421i)8-s + (−1.71 + 2.46i)9-s + (−1.41 + 0.0704i)10-s + (−3.19 + 1.84i)11-s + (−3.45 + 0.198i)12-s + (2.94 + 1.69i)13-s + (5.91 + 3.03i)14-s + (−1.73 − 0.0730i)15-s + (−2.64 − 2.99i)16-s − 3.19i·17-s + ⋯
L(s)  = 1  + (0.542 + 0.840i)2-s + (0.463 + 0.886i)3-s + (−0.411 + 0.911i)4-s + (−0.223 + 0.387i)5-s + (−0.493 + 0.869i)6-s + (1.53 − 0.888i)7-s + (−0.988 + 0.148i)8-s + (−0.571 + 0.820i)9-s + (−0.446 + 0.0222i)10-s + (−0.963 + 0.556i)11-s + (−0.998 + 0.0573i)12-s + (0.816 + 0.471i)13-s + (1.58 + 0.810i)14-s + (−0.446 − 0.0188i)15-s + (−0.661 − 0.749i)16-s − 0.776i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.691 - 0.722i$
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: \((2,\ 360,\ (\ :1/2),\ -0.691 - 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.783952 + 1.83466i\)
\(L(\frac12)\) \(\approx\) \(0.783952 + 1.83466i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.767 - 1.18i)T \)
3 \( 1 + (-0.801 - 1.53i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (-4.07 + 2.35i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.19 - 1.84i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.94 - 1.69i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.19iT - 17T^{2} \)
19 \( 1 + 0.184T + 19T^{2} \)
23 \( 1 + (-1.68 + 2.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.71 + 8.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.31 - 0.761i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.21iT - 37T^{2} \)
41 \( 1 + (-7.97 - 4.60i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.05 - 8.75i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.20 - 5.54i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.652T + 53T^{2} \)
59 \( 1 + (-7.25 - 4.19i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.59 + 3.23i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.725 - 1.25i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 + 15.3T + 73T^{2} \)
79 \( 1 + (-2.19 + 1.26i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.86 - 2.23i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.32iT - 89T^{2} \)
97 \( 1 + (8.59 + 14.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.47861123579908729137390902568, −11.04394013822158119224549151883, −9.908416762500621379038381643270, −8.733795610043998049422089159684, −7.83606883853419955713693617195, −7.36394645235522552418524028444, −5.77293745606014652081992631989, −4.57324926315804983407068559899, −4.20676670912699521588700146042, −2.64664009423273056270170114370, 1.30360709971942377791351943313, 2.42719807665655559476997129300, 3.73845084268036499589574644148, 5.29629968679732404999648831425, 5.78996929492371878844204907890, 7.53828888786461258011162937560, 8.572041755691378241547650963054, 8.868794025311432994249878467619, 10.61142603592035466545126547220, 11.27460470094010306425950420195

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