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Properties

Label 2-360-120.53-c1-0-9
Degree $2$
Conductor $360$
Sign $0.251 - 0.967i$
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  + (−0.473 + 1.33i)2-s + (−1.55 − 1.26i)4-s + (1.45 + 1.69i)5-s + (1.53 − 1.53i)7-s + (2.41 − 1.47i)8-s + (−2.95 + 1.13i)10-s + 2.72·11-s + (0.857 − 0.857i)13-s + (1.31 + 2.76i)14-s + (0.818 + 3.91i)16-s + (2.55 + 2.55i)17-s − 3.54·19-s + (−0.113 − 4.47i)20-s + (−1.28 + 3.63i)22-s + (−0.626 + 0.626i)23-s + ⋯
L(s)  = 1  + (−0.334 + 0.942i)2-s + (−0.776 − 0.630i)4-s + (0.650 + 0.759i)5-s + (0.578 − 0.578i)7-s + (0.853 − 0.520i)8-s + (−0.933 + 0.358i)10-s + 0.821·11-s + (0.237 − 0.237i)13-s + (0.351 + 0.738i)14-s + (0.204 + 0.978i)16-s + (0.619 + 0.619i)17-s − 0.812·19-s + (−0.0254 − 0.999i)20-s + (−0.274 + 0.774i)22-s + (−0.130 + 0.130i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.251 - 0.967i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.251 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02457 + 0.792700i\)
\(L(\frac12)\) \(\approx\) \(1.02457 + 0.792700i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.473 - 1.33i)T \)
3 \( 1 \)
5 \( 1 + (-1.45 - 1.69i)T \)
good7 \( 1 + (-1.53 + 1.53i)T - 7iT^{2} \)
11 \( 1 - 2.72T + 11T^{2} \)
13 \( 1 + (-0.857 + 0.857i)T - 13iT^{2} \)
17 \( 1 + (-2.55 - 2.55i)T + 17iT^{2} \)
19 \( 1 + 3.54T + 19T^{2} \)
23 \( 1 + (0.626 - 0.626i)T - 23iT^{2} \)
29 \( 1 - 5.12iT - 29T^{2} \)
31 \( 1 - 7.89T + 31T^{2} \)
37 \( 1 + (4.21 + 4.21i)T + 37iT^{2} \)
41 \( 1 - 12.4iT - 41T^{2} \)
43 \( 1 + (-5.67 + 5.67i)T - 43iT^{2} \)
47 \( 1 + (9.45 + 9.45i)T + 47iT^{2} \)
53 \( 1 + (6.46 + 6.46i)T + 53iT^{2} \)
59 \( 1 + 2.51iT - 59T^{2} \)
61 \( 1 + 9.49iT - 61T^{2} \)
67 \( 1 + (-9.91 - 9.91i)T + 67iT^{2} \)
71 \( 1 - 2.19iT - 71T^{2} \)
73 \( 1 + (5.71 + 5.71i)T + 73iT^{2} \)
79 \( 1 + 12.7iT - 79T^{2} \)
83 \( 1 + (3.58 + 3.58i)T + 83iT^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 + (1.29 - 1.29i)T - 97iT^{2} \)
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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

−11.34007864886512833715379418446, −10.46118404073064157889278062136, −9.830492173143899820861584712913, −8.699681524118870143119804935410, −7.82985019775255639138346080098, −6.76969275564290286518626552317, −6.14150629474063518942628290237, −4.92999810818858339485702278479, −3.65340844576073601755125590738, −1.54072316143711982322900013436, 1.28270719605535487965185863854, 2.49959852062694329510674496872, 4.15148735461421931864076358009, 5.07809591274727682486585633844, 6.29976758612026043986589884068, 7.933277793186658596398595302111, 8.722423709933537238997330812200, 9.415395914773666729719334960886, 10.24560549379053070702314506218, 11.39607754721079929269210229242

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