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Properties

Label 2-360-72.59-c1-0-6
Degree $2$
Conductor $360$
Sign $0.538 - 0.842i$
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  + (−1.29 − 0.574i)2-s + (−0.857 − 1.50i)3-s + (1.33 + 1.48i)4-s + (−0.5 + 0.866i)5-s + (0.243 + 2.43i)6-s + (0.661 − 0.382i)7-s + (−0.877 − 2.68i)8-s + (−1.52 + 2.58i)9-s + (1.14 − 0.831i)10-s + (−4.25 + 2.45i)11-s + (1.08 − 3.28i)12-s + (2.38 + 1.37i)13-s + (−1.07 + 0.113i)14-s + (1.73 + 0.00945i)15-s + (−0.411 + 3.97i)16-s + 4.18i·17-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (−0.495 − 0.868i)3-s + (0.669 + 0.742i)4-s + (−0.223 + 0.387i)5-s + (0.0995 + 0.995i)6-s + (0.250 − 0.144i)7-s + (−0.310 − 0.950i)8-s + (−0.509 + 0.860i)9-s + (0.361 − 0.263i)10-s + (−1.28 + 0.741i)11-s + (0.313 − 0.949i)12-s + (0.661 + 0.381i)13-s + (−0.287 + 0.0303i)14-s + (0.447 + 0.00244i)15-s + (−0.102 + 0.994i)16-s + 1.01i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.538 - 0.842i$
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: \((2,\ 360,\ (\ :1/2),\ 0.538 - 0.842i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.396688 + 0.217403i\)
\(L(\frac12)\) \(\approx\) \(0.396688 + 0.217403i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 + 0.574i)T \)
3 \( 1 + (0.857 + 1.50i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (-0.661 + 0.382i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.25 - 2.45i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.38 - 1.37i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.18iT - 17T^{2} \)
19 \( 1 + 5.61T + 19T^{2} \)
23 \( 1 + (-0.420 + 0.727i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.03 - 3.53i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-9.36 - 5.40i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.95iT - 37T^{2} \)
41 \( 1 + (-4.74 - 2.73i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.563 + 0.976i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.53 + 4.39i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 12.8T + 53T^{2} \)
59 \( 1 + (2.54 + 1.46i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.15 + 1.24i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.544 + 0.942i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + 6.10T + 73T^{2} \)
79 \( 1 + (14.1 - 8.14i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.75 - 1.58i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.54iT - 89T^{2} \)
97 \( 1 + (9.53 + 16.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.33928728299911579639570380841, −10.73391608676027736641754173561, −10.07691574867303329570275028098, −8.420380019550209905268544497764, −8.068810703759979616694976810664, −6.92657510419642481595311559219, −6.25140014897753680416931982968, −4.57819666059810517796066993756, −2.82654414986066943026753484357, −1.60206270938607318171047104767, 0.43801971005749221512424100458, 2.78854883435045604732505792545, 4.55416073662021534210203685179, 5.55743649006868118051894200021, 6.34695838725679126932940026681, 7.88571943297581426322406506151, 8.485333176311733331114918729327, 9.451672355848651740523154800585, 10.36929517518223559187393533338, 11.06157146285164199935788889413

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