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Properties

Label 2-180-36.23-c1-0-11
Degree $2$
Conductor $180$
Sign $0.243 - 0.969i$
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  + (0.752 + 1.19i)2-s + (1.67 + 0.425i)3-s + (−0.866 + 1.80i)4-s + (−0.866 − 0.5i)5-s + (0.755 + 2.33i)6-s + (1.87 − 1.08i)7-s + (−2.81 + 0.320i)8-s + (2.63 + 1.42i)9-s + (−0.0535 − 1.41i)10-s + (−1.44 − 2.50i)11-s + (−2.22 + 2.65i)12-s + (−3.30 + 5.72i)13-s + (2.70 + 1.42i)14-s + (−1.24 − 1.20i)15-s + (−2.49 − 3.12i)16-s − 5.08i·17-s + ⋯
L(s)  = 1  + (0.532 + 0.846i)2-s + (0.969 + 0.245i)3-s + (−0.433 + 0.901i)4-s + (−0.387 − 0.223i)5-s + (0.308 + 0.951i)6-s + (0.708 − 0.409i)7-s + (−0.993 + 0.113i)8-s + (0.879 + 0.475i)9-s + (−0.0169 − 0.446i)10-s + (−0.436 − 0.755i)11-s + (−0.641 + 0.767i)12-s + (−0.916 + 1.58i)13-s + (0.723 + 0.381i)14-s + (−0.320 − 0.311i)15-s + (−0.624 − 0.780i)16-s − 1.23i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.243 - 0.969i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.243 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42348 + 1.11063i\)
\(L(\frac12)\) \(\approx\) \(1.42348 + 1.11063i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.752 - 1.19i)T \)
3 \( 1 + (-1.67 - 0.425i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
good7 \( 1 + (-1.87 + 1.08i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.44 + 2.50i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.30 - 5.72i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.08iT - 17T^{2} \)
19 \( 1 + 4.66iT - 19T^{2} \)
23 \( 1 + (-1.44 + 2.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.99 - 3.46i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.51 - 2.60i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.58T + 37T^{2} \)
41 \( 1 + (4.18 + 2.41i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.81 + 2.20i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.00969 + 0.0167i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.21iT - 53T^{2} \)
59 \( 1 + (4.02 - 6.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.321 + 0.556i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.19 + 4.73i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.17T + 71T^{2} \)
73 \( 1 - 2.32T + 73T^{2} \)
79 \( 1 + (14.6 - 8.48i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.950 - 1.64i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.72iT - 89T^{2} \)
97 \( 1 + (-5.78 - 10.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.31177295686094388358789230779, −12.05800969581971241167533115114, −11.06081151965261103825412505874, −9.375197385639189151718324851308, −8.725233813870465318588713002514, −7.59560361710766594861269358482, −6.96149150441483469378453181533, −4.98090733623186572101031866504, −4.31417476776804827226404136156, −2.78620197085746732649320533917, 1.93979713922192339796203168197, 3.17944146564426709987227164451, 4.47073597554836566044287003588, 5.79832707952199609073355936910, 7.61395088298345181250754105551, 8.298436012619391685490565016237, 9.749539418340600072930702462201, 10.38535950771150522579629283561, 11.67574575433891449904973781929, 12.65192595340686111262867492262

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