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Properties

Label 2-360-72.59-c1-0-3
Degree $2$
Conductor $360$
Sign $-0.921 - 0.389i$
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  + (−1.10 + 0.880i)2-s + (0.424 − 1.67i)3-s + (0.450 − 1.94i)4-s + (−0.5 + 0.866i)5-s + (1.00 + 2.23i)6-s + (−3.88 + 2.24i)7-s + (1.21 + 2.55i)8-s + (−2.63 − 1.42i)9-s + (−0.208 − 1.39i)10-s + (1.05 − 0.606i)11-s + (−3.08 − 1.58i)12-s + (−1.71 − 0.987i)13-s + (2.32 − 5.90i)14-s + (1.24 + 1.20i)15-s + (−3.59 − 1.75i)16-s + 5.28i·17-s + ⋯
L(s)  = 1  + (−0.782 + 0.622i)2-s + (0.245 − 0.969i)3-s + (0.225 − 0.974i)4-s + (−0.223 + 0.387i)5-s + (0.411 + 0.911i)6-s + (−1.46 + 0.847i)7-s + (0.430 + 0.902i)8-s + (−0.879 − 0.475i)9-s + (−0.0660 − 0.442i)10-s + (0.316 − 0.182i)11-s + (−0.889 − 0.457i)12-s + (−0.474 − 0.273i)13-s + (0.621 − 1.57i)14-s + (0.320 + 0.311i)15-s + (−0.898 − 0.438i)16-s + 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.921 - 0.389i$
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: \((2,\ 360,\ (\ :1/2),\ -0.921 - 0.389i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0389965 + 0.192597i\)
\(L(\frac12)\) \(\approx\) \(0.0389965 + 0.192597i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.10 - 0.880i)T \)
3 \( 1 + (-0.424 + 1.67i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (3.88 - 2.24i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.05 + 0.606i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.71 + 0.987i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.28iT - 17T^{2} \)
19 \( 1 + 4.86T + 19T^{2} \)
23 \( 1 + (1.40 - 2.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.20 - 7.28i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.04 + 3.49i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.71iT - 37T^{2} \)
41 \( 1 + (5.30 + 3.06i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.17 - 2.04i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.97 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.739T + 53T^{2} \)
59 \( 1 + (-1.70 - 0.986i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.81 + 2.77i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.58 + 2.74i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.40T + 71T^{2} \)
73 \( 1 + 6.69T + 73T^{2} \)
79 \( 1 + (-11.2 + 6.49i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.00 - 2.89i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.71iT - 89T^{2} \)
97 \( 1 + (-8.61 - 14.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.93688066696054097914386685006, −10.74476011092075812796264275329, −9.793962861771291009096642204593, −8.826398728831890429798462513230, −8.203524873785643109766416611049, −6.95724363618708101130027422704, −6.43511937964532221059635952818, −5.60815333621776115203225710470, −3.36464290949910145338776249299, −2.03155042987103453081508940355, 0.15528127615440519813777470247, 2.63802169179543586588466179407, 3.75799883715054384111397932974, 4.57301716610455687928270401780, 6.44622751565950791254236531451, 7.44427144333397897055719357091, 8.620038873138905508246429796212, 9.473301124213333925503936718640, 9.939079256381928920552659771110, 10.77015997552359458211011793148

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