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Properties

Label 2-180-180.119-c1-0-7
Degree $2$
Conductor $180$
Sign $-0.552 - 0.833i$
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  + (0.0443 + 1.41i)2-s + (1.66 + 0.461i)3-s + (−1.99 + 0.125i)4-s + (−1.69 + 1.46i)5-s + (−0.578 + 2.38i)6-s + (−0.667 + 1.15i)7-s + (−0.265 − 2.81i)8-s + (2.57 + 1.54i)9-s + (−2.14 − 2.32i)10-s + (−2.18 + 3.78i)11-s + (−3.39 − 0.711i)12-s + (3.56 − 2.05i)13-s + (−1.66 − 0.892i)14-s + (−3.49 + 1.65i)15-s + (3.96 − 0.500i)16-s + 6.45·17-s + ⋯
L(s)  = 1  + (0.0313 + 0.999i)2-s + (0.963 + 0.266i)3-s + (−0.998 + 0.0626i)4-s + (−0.756 + 0.653i)5-s + (−0.236 + 0.971i)6-s + (−0.252 + 0.436i)7-s + (−0.0939 − 0.995i)8-s + (0.858 + 0.513i)9-s + (−0.677 − 0.735i)10-s + (−0.659 + 1.14i)11-s + (−0.978 − 0.205i)12-s + (0.987 − 0.570i)13-s + (−0.444 − 0.238i)14-s + (−0.903 + 0.428i)15-s + (0.992 − 0.125i)16-s + 1.56·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.552 - 0.833i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ -0.552 - 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.601835 + 1.12023i\)
\(L(\frac12)\) \(\approx\) \(0.601835 + 1.12023i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0443 - 1.41i)T \)
3 \( 1 + (-1.66 - 0.461i)T \)
5 \( 1 + (1.69 - 1.46i)T \)
good7 \( 1 + (0.667 - 1.15i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.18 - 3.78i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.56 + 2.05i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.45T + 17T^{2} \)
19 \( 1 + 5.84iT - 19T^{2} \)
23 \( 1 + (0.0875 - 0.0505i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.53 + 2.61i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.18 - 2.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.24iT - 37T^{2} \)
41 \( 1 + (-3.50 + 2.02i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.92 + 3.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.00 - 1.73i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.77T + 53T^{2} \)
59 \( 1 + (-1.37 - 2.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.04 + 1.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.216 - 0.374i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.41T + 71T^{2} \)
73 \( 1 - 7.28iT - 73T^{2} \)
79 \( 1 + (2.58 + 1.49i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.6 + 6.70i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.97iT - 89T^{2} \)
97 \( 1 + (-2.08 - 1.20i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.16353478604582109114388845780, −12.41458219203874847609173108127, −10.73494778904446331917168383498, −9.756181053631023075356176631005, −8.759979309517763828486403023209, −7.70288312649755919746458539938, −7.18391304264471236045486961613, −5.57620529476379064870750502932, −4.17281777479350488744826676872, −3.01419447876660822049787782313, 1.25937226434728158203338170528, 3.33095303147848002211954231918, 3.93789875399484171917226888159, 5.66455351217872499528670480903, 7.70748294441521085703689544205, 8.338137169909257828860305202095, 9.273991231310350205746916098385, 10.35505595468921426183857608423, 11.43029569704548564963048971440, 12.44666520756399568710708479895

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