Properties

Label 2-360-72.59-c1-0-20
Degree $2$
Conductor $360$
Sign $0.781 - 0.624i$
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  + (1.39 + 0.254i)2-s + (−1.73 − 0.00397i)3-s + (1.87 + 0.708i)4-s + (−0.5 + 0.866i)5-s + (−2.40 − 0.446i)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 + (−3.23 − 1.23i)12-s + (3.30 + 1.90i)13-s + (3.69 − 1.31i)14-s + (0.869 − 1.49i)15-s + (2.99 + 2.64i)16-s + 3.43i·17-s + ⋯
L(s)  = 1  + (0.983 + 0.179i)2-s + (−0.999 − 0.00229i)3-s + (0.935 + 0.354i)4-s + (−0.223 + 0.387i)5-s + (−0.983 − 0.182i)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 + (−0.934 − 0.356i)12-s + (0.916 + 0.529i)13-s + (0.988 − 0.352i)14-s + (0.224 − 0.386i)15-s + (0.749 + 0.662i)16-s + 0.833i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.781 - 0.624i$
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: \((2,\ 360,\ (\ :1/2),\ 0.781 - 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82733 + 0.640613i\)
\(L(\frac12)\) \(\approx\) \(1.82733 + 0.640613i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.254i)T \)
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

−11.62387282818847522398704964593, −10.69966640139548720881674002937, −10.44282649206031192840044689704, −8.421299535793685337965353762981, −7.37340494565546167248257624094, −6.67369642032780391957906274641, −5.55709016532224072617842858147, −4.67184278478721903155764786634, −3.73369615675739703540683861611, −1.80695698940572771515728198403, 1.36178310319625697331854270538, 3.18748554891766997325150351000, 4.70994069710893458813618830653, 5.28142063691249737736123974655, 6.11207743154470492684925942429, 7.38409951045575146519081944567, 8.331572200383176558025424737936, 9.864714998629303071449368502276, 10.91236385661302866846901712711, 11.47368523109657683730469927427

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