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Properties

Label 2-360-120.53-c1-0-13
Degree $2$
Conductor $360$
Sign $0.0531 + 0.998i$
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  + (−0.319 − 1.37i)2-s + (−1.79 + 0.879i)4-s + (−1.42 + 1.72i)5-s + (3.11 − 3.11i)7-s + (1.78 + 2.19i)8-s + (2.82 + 1.41i)10-s + 1.17·11-s + (2.15 − 2.15i)13-s + (−5.29 − 3.30i)14-s + (2.45 − 3.15i)16-s + (−1.33 − 1.33i)17-s + 0.322·19-s + (1.04 − 4.34i)20-s + (−0.375 − 1.62i)22-s + (4.71 − 4.71i)23-s + ⋯
L(s)  = 1  + (−0.225 − 0.974i)2-s + (−0.898 + 0.439i)4-s + (−0.637 + 0.770i)5-s + (1.17 − 1.17i)7-s + (0.630 + 0.775i)8-s + (0.894 + 0.447i)10-s + 0.355·11-s + (0.596 − 0.596i)13-s + (−1.41 − 0.882i)14-s + (0.613 − 0.789i)16-s + (−0.323 − 0.323i)17-s + 0.0739·19-s + (0.234 − 0.972i)20-s + (−0.0801 − 0.346i)22-s + (0.982 − 0.982i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0531 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0531 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.0531 + 0.998i$
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: \((2,\ 360,\ (\ :1/2),\ 0.0531 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.818546 - 0.776114i\)
\(L(\frac12)\) \(\approx\) \(0.818546 - 0.776114i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.319 + 1.37i)T \)
3 \( 1 \)
5 \( 1 + (1.42 - 1.72i)T \)
good7 \( 1 + (-3.11 + 3.11i)T - 7iT^{2} \)
11 \( 1 - 1.17T + 11T^{2} \)
13 \( 1 + (-2.15 + 2.15i)T - 13iT^{2} \)
17 \( 1 + (1.33 + 1.33i)T + 17iT^{2} \)
19 \( 1 - 0.322T + 19T^{2} \)
23 \( 1 + (-4.71 + 4.71i)T - 23iT^{2} \)
29 \( 1 + 6.63iT - 29T^{2} \)
31 \( 1 + 0.0675T + 31T^{2} \)
37 \( 1 + (-7.60 - 7.60i)T + 37iT^{2} \)
41 \( 1 - 3.19iT - 41T^{2} \)
43 \( 1 + (6.70 - 6.70i)T - 43iT^{2} \)
47 \( 1 + (-7.34 - 7.34i)T + 47iT^{2} \)
53 \( 1 + (5.73 + 5.73i)T + 53iT^{2} \)
59 \( 1 - 8.68iT - 59T^{2} \)
61 \( 1 + 12.5iT - 61T^{2} \)
67 \( 1 + (-1.87 - 1.87i)T + 67iT^{2} \)
71 \( 1 + 4.18iT - 71T^{2} \)
73 \( 1 + (-3.97 - 3.97i)T + 73iT^{2} \)
79 \( 1 - 9.66iT - 79T^{2} \)
83 \( 1 + (-0.585 - 0.585i)T + 83iT^{2} \)
89 \( 1 - 0.557T + 89T^{2} \)
97 \( 1 + (10.5 - 10.5i)T - 97iT^{2} \)
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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

−11.14920396378665741488157892399, −10.63750194106072283715901983584, −9.656023700626210210551965524231, −8.236101027169887479296892464118, −7.83379053850829496841535998555, −6.61495051455590887645647713882, −4.76380362238326480858469287931, −4.02437509698345146079808359791, −2.80425711396663126933167431147, −1.02291455702508365864028811353, 1.51653840577121012968158326695, 3.93598119407182591185630240386, 5.00788073631247102566336724273, 5.69886514568414467090876665575, 7.06355246234985475516116594111, 8.027357974559665068839866029705, 8.922865693771669245253117334678, 9.108626899371479898715867980321, 10.85964020626912122251017799815, 11.67241202116750779855413155316

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