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Properties

Label 2-180-36.11-c1-0-8
Degree $2$
Conductor $180$
Sign $0.952 + 0.305i$
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  + (0.0785 − 1.41i)2-s + (0.427 + 1.67i)3-s + (−1.98 − 0.221i)4-s + (0.866 − 0.5i)5-s + (2.40 − 0.472i)6-s + (4.06 + 2.34i)7-s + (−0.469 + 2.78i)8-s + (−2.63 + 1.43i)9-s + (−0.638 − 1.26i)10-s + (1.34 − 2.32i)11-s + (−0.477 − 3.43i)12-s + (−1.04 − 1.80i)13-s + (3.63 − 5.56i)14-s + (1.20 + 1.23i)15-s + (3.90 + 0.881i)16-s − 3.78i·17-s + ⋯
L(s)  = 1  + (0.0555 − 0.998i)2-s + (0.246 + 0.969i)3-s + (−0.993 − 0.110i)4-s + (0.387 − 0.223i)5-s + (0.981 − 0.192i)6-s + (1.53 + 0.888i)7-s + (−0.165 + 0.986i)8-s + (−0.878 + 0.478i)9-s + (−0.201 − 0.399i)10-s + (0.404 − 0.701i)11-s + (−0.137 − 0.990i)12-s + (−0.289 − 0.501i)13-s + (0.972 − 1.48i)14-s + (0.312 + 0.320i)15-s + (0.975 + 0.220i)16-s − 0.918i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.952 + 0.305i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.952 + 0.305i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32721 - 0.207518i\)
\(L(\frac12)\) \(\approx\) \(1.32721 - 0.207518i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0785 + 1.41i)T \)
3 \( 1 + (-0.427 - 1.67i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
good7 \( 1 + (-4.06 - 2.34i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.34 + 2.32i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.04 + 1.80i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.78iT - 17T^{2} \)
19 \( 1 - 3.78iT - 19T^{2} \)
23 \( 1 + (0.163 + 0.282i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.75 + 1.01i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.70 - 4.44i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.76T + 37T^{2} \)
41 \( 1 + (2.34 - 1.35i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.18 + 3.57i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.465 + 0.805i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.41iT - 53T^{2} \)
59 \( 1 + (-1.58 - 2.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.65 + 11.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.62 + 2.66i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.2T + 71T^{2} \)
73 \( 1 - 13.7T + 73T^{2} \)
79 \( 1 + (-7.94 - 4.58i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.11 + 5.39i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.99iT - 89T^{2} \)
97 \( 1 + (-0.0157 + 0.0272i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.29284716012193612927605000332, −11.48648490466079465531982024706, −10.76137872464994353927057278239, −9.678616494770182649584723273403, −8.778121975529569599811892733083, −8.133916511601063114116337791400, −5.47440670928471031301797128844, −5.01160214913301973457873465591, −3.51004352906008492036537459428, −2.02424849648677322028985117770, 1.67440748926646656339123129599, 4.09963294106933335604578654631, 5.36327718726314375377914080548, 6.76849670121717508699959683219, 7.39033941822783130443999292421, 8.312549968583758930419617129137, 9.344393677937618361675799472140, 10.76824531191860793466506243566, 11.91231339889024848752625186390, 13.08841691445465552441754001476

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