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Properties

Label 2-360-120.53-c1-0-18
Degree $2$
Conductor $360$
Sign $0.923 - 0.383i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.923 - 0.383i$
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: \((2,\ 360,\ (\ :1/2),\ 0.923 - 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60723 + 0.319986i\)
\(L(\frac12)\) \(\approx\) \(1.60723 + 0.319986i\)
\(L(\frac{3}{2})\) 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

−11.55398410190204956846457671855, −10.39135720886510369246234783387, −9.578038936614477009301042994345, −8.178146264291876775992894997923, −8.035158529919639231075349374981, −6.70102989419689778912634479383, −5.51616230445977368143754515896, −4.82167217062477475936057902944, −3.70129260153755118566435311121, −1.27574799560777029423743590499, 1.87729511163726794958381531330, 2.69174958545185976879351910684, 4.26649562015962978658346157338, 5.44865522395390840247628907276, 6.19221847890395107473234679620, 7.904292305235517770087550451619, 8.807149853776468284267120774914, 9.719326887995810413059667597464, 10.62395973089674103501857773767, 11.47623872718097326997563732956

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