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Properties

Label 2-180-180.59-c1-0-10
Degree $2$
Conductor $180$
Sign $0.192 - 0.981i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
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  + (0.356 + 1.36i)2-s + (1.30 − 1.14i)3-s + (−1.74 + 0.974i)4-s + (−0.493 + 2.18i)5-s + (2.02 + 1.37i)6-s + (1.65 + 2.87i)7-s + (−1.95 − 2.04i)8-s + (0.391 − 2.97i)9-s + (−3.16 + 0.100i)10-s + (−1.17 − 2.04i)11-s + (−1.16 + 3.26i)12-s + (3.81 + 2.20i)13-s + (−3.33 + 3.29i)14-s + (1.84 + 3.40i)15-s + (2.10 − 3.40i)16-s + 0.889·17-s + ⋯
L(s)  = 1  + (0.251 + 0.967i)2-s + (0.751 − 0.659i)3-s + (−0.873 + 0.487i)4-s + (−0.220 + 0.975i)5-s + (0.827 + 0.561i)6-s + (0.626 + 1.08i)7-s + (−0.691 − 0.722i)8-s + (0.130 − 0.991i)9-s + (−0.999 + 0.0318i)10-s + (−0.355 − 0.615i)11-s + (−0.335 + 0.942i)12-s + (1.05 + 0.610i)13-s + (−0.892 + 0.879i)14-s + (0.477 + 0.878i)15-s + (0.525 − 0.851i)16-s + 0.215·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.192 - 0.981i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.192 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15829 + 0.953410i\)
\(L(\frac12)\) \(\approx\) \(1.15829 + 0.953410i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.356 - 1.36i)T \)
3 \( 1 + (-1.30 + 1.14i)T \)
5 \( 1 + (0.493 - 2.18i)T \)
good7 \( 1 + (-1.65 - 2.87i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.17 + 2.04i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.81 - 2.20i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.889T + 17T^{2} \)
19 \( 1 + 3.03iT - 19T^{2} \)
23 \( 1 + (6.29 + 3.63i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.03 + 0.598i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.205 + 0.118i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 11.4iT - 37T^{2} \)
41 \( 1 + (-2.45 - 1.41i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.17 - 7.23i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.55 - 3.78i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.772T + 53T^{2} \)
59 \( 1 + (1.06 - 1.84i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.91 - 3.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.17 + 5.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 - 6.10iT - 73T^{2} \)
79 \( 1 + (8.94 - 5.16i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.25 - 1.29i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.68iT - 89T^{2} \)
97 \( 1 + (5.34 - 3.08i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.14925080222332933701415773626, −12.10579311585444557255529138213, −11.10961997273289834241015005696, −9.412291004496386082445797228766, −8.452823012643949151444870893606, −7.82917606805805212771433103300, −6.59198073989081554246979272365, −5.82165438374225184768861089476, −3.96775606933829012877886786784, −2.59204150829525482254145881153, 1.57966932763302807437229963216, 3.59503330283491555798765291809, 4.38177638218836849193499388264, 5.43226213486669071093595858883, 7.87930249734505062732541832824, 8.430746903947806395011808394767, 9.742198719401689326117867340833, 10.35802934563283401841403870395, 11.36641031082091174486341788025, 12.51308890924970456964399834263

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