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Properties

Label 2-180-9.4-c1-0-3
Degree $2$
Conductor $180$
Sign $0.777 + 0.629i$
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  + (1.62 − 0.606i)3-s + (0.5 − 0.866i)5-s + (−2.05 − 3.55i)7-s + (2.26 − 1.96i)9-s + (1.90 + 3.30i)11-s + (−2.90 + 5.03i)13-s + (0.285 − 1.70i)15-s + 3.81·17-s − 1.81·19-s + (−5.48 − 4.51i)21-s + (−1.05 + 1.81i)23-s + (−0.499 − 0.866i)25-s + (2.48 − 4.56i)27-s + (3.60 + 6.23i)29-s + (0.908 − 1.57i)31-s + ⋯
L(s)  = 1  + (0.936 − 0.350i)3-s + (0.223 − 0.387i)5-s + (−0.774 − 1.34i)7-s + (0.754 − 0.655i)9-s + (0.575 + 0.996i)11-s + (−0.806 + 1.39i)13-s + (0.0738 − 0.441i)15-s + 0.925·17-s − 0.416·19-s + (−1.19 − 0.985i)21-s + (−0.219 + 0.379i)23-s + (−0.0999 − 0.173i)25-s + (0.477 − 0.878i)27-s + (0.668 + 1.15i)29-s + (0.163 − 0.282i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.777 + 0.629i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.777 + 0.629i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40762 - 0.498581i\)
\(L(\frac12)\) \(\approx\) \(1.40762 - 0.498581i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.62 + 0.606i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (2.05 + 3.55i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.90 - 3.30i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.90 - 5.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.81T + 17T^{2} \)
19 \( 1 + 1.81T + 19T^{2} \)
23 \( 1 + (1.05 - 1.81i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.60 - 6.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.908 + 1.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.01T + 37T^{2} \)
41 \( 1 + (5.50 - 9.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.90 + 5.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.95 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.20T + 53T^{2} \)
59 \( 1 + (-2.10 + 3.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.50 - 2.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.85 - 3.21i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.01T + 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.94 - 3.37i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (6.10 + 10.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.66290972737933379711288180818, −11.89528560969671722985046104536, −10.03785906066832924770735467008, −9.738874686110912607428473985091, −8.544434220802279174481400605809, −7.14332255260233714086617465413, −6.78854606336726132168030007934, −4.60604271600267756246747591714, −3.55215782553355561492622503485, −1.68651022375172412016611166393, 2.58827286463873851948168485648, 3.41428187158265913211455698358, 5.31838998301792099827292424215, 6.38901620022697987004392379276, 7.920202809599739674390386013262, 8.760413288715262846565533660475, 9.741935931729987978059252649686, 10.46892668500444656351483711740, 11.98577012667256289690186877210, 12.77876124616036415331055695788

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