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Properties

Label 2-360-72.59-c1-0-1
Degree $2$
Conductor $360$
Sign $-0.187 - 0.982i$
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  + (0.475 − 1.33i)2-s + (−1.73 − 0.00397i)3-s + (−1.54 − 1.26i)4-s + (0.5 − 0.866i)5-s + (−0.828 + 2.30i)6-s + (−2.40 + 1.38i)7-s + (−2.42 + 1.46i)8-s + (2.99 + 0.0137i)9-s + (−0.915 − 1.07i)10-s + (−2.70 + 1.56i)11-s + (2.67 + 2.19i)12-s + (−3.30 − 1.90i)13-s + (0.706 + 3.86i)14-s + (−0.869 + 1.49i)15-s + (0.795 + 3.92i)16-s + 3.43i·17-s + ⋯
L(s)  = 1  + (0.336 − 0.941i)2-s + (−0.999 − 0.00229i)3-s + (−0.774 − 0.632i)4-s + (0.223 − 0.387i)5-s + (−0.338 + 0.941i)6-s + (−0.908 + 0.524i)7-s + (−0.856 + 0.516i)8-s + (0.999 + 0.00459i)9-s + (−0.289 − 0.340i)10-s + (−0.815 + 0.470i)11-s + (0.772 + 0.634i)12-s + (−0.916 − 0.529i)13-s + (0.188 + 1.03i)14-s + (−0.224 + 0.386i)15-s + (0.198 + 0.980i)16-s + 0.833i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.187 - 0.982i$
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: \((2,\ 360,\ (\ :1/2),\ -0.187 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0245093 + 0.0296451i\)
\(L(\frac12)\) \(\approx\) \(0.0245093 + 0.0296451i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.475 + 1.33i)T \)
3 \( 1 + (1.73 + 0.00397i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (2.40 - 1.38i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.70 - 1.56i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.30 + 1.90i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.43iT - 17T^{2} \)
19 \( 1 - 1.31T + 19T^{2} \)
23 \( 1 + (2.13 - 3.69i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.266 + 0.461i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.413 - 0.238i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.26iT - 37T^{2} \)
41 \( 1 + (9.97 + 5.76i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.25 + 10.8i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.52 - 2.64i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 14.2T + 53T^{2} \)
59 \( 1 + (8.89 + 5.13i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.7 + 6.18i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.47 + 6.02i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.88T + 71T^{2} \)
73 \( 1 - 7.92T + 73T^{2} \)
79 \( 1 + (-0.187 + 0.108i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.86 - 4.54i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.82iT - 89T^{2} \)
97 \( 1 + (4.20 + 7.27i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.91402528576966032641092205047, −10.76566530917498313622488304952, −9.981658982134848669886364091616, −9.503236692745874464980510520929, −8.056580250875151683956085285773, −6.59165989475109921460170089164, −5.51234263805674235731014304889, −4.96056346474381878810052920767, −3.49813973597626546156474544186, −1.98221954328759472762722938553, 0.02526829069747722244810609852, 3.07431922563333766968613330949, 4.50448139574453256192613912142, 5.40188618033539884265525001543, 6.49585977852959981330759275690, 6.98784534667559258091866002934, 8.022403309844264366075649271378, 9.590780930819285547954842373345, 10.05232152271235930588309962790, 11.27883001511238001543528122952

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