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Properties

Label 2-180-180.119-c1-0-2
Degree $2$
Conductor $180$
Sign $-0.999 + 0.0433i$
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  + (0.745 + 1.20i)2-s + (−1.62 + 0.589i)3-s + (−0.888 + 1.79i)4-s + (−1.91 − 1.14i)5-s + (−1.92 − 1.51i)6-s + (−1.44 + 2.50i)7-s + (−2.81 + 0.269i)8-s + (2.30 − 1.91i)9-s + (−0.0493 − 3.16i)10-s + (−0.395 + 0.684i)11-s + (0.390 − 3.44i)12-s + (−4.04 + 2.33i)13-s + (−4.08 + 0.129i)14-s + (3.80 + 0.741i)15-s + (−2.42 − 3.18i)16-s + 5.89·17-s + ⋯
L(s)  = 1  + (0.527 + 0.849i)2-s + (−0.940 + 0.340i)3-s + (−0.444 + 0.896i)4-s + (−0.857 − 0.513i)5-s + (−0.784 − 0.619i)6-s + (−0.546 + 0.945i)7-s + (−0.995 + 0.0951i)8-s + (0.768 − 0.639i)9-s + (−0.0156 − 0.999i)10-s + (−0.119 + 0.206i)11-s + (0.112 − 0.993i)12-s + (−1.12 + 0.647i)13-s + (−1.09 + 0.0346i)14-s + (0.981 + 0.191i)15-s + (−0.605 − 0.795i)16-s + 1.42·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0433i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.999 + 0.0433i$
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: \((2,\ 180,\ (\ :1/2),\ -0.999 + 0.0433i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0134912 - 0.622614i\)
\(L(\frac12)\) \(\approx\) \(0.0134912 - 0.622614i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.745 - 1.20i)T \)
3 \( 1 + (1.62 - 0.589i)T \)
5 \( 1 + (1.91 + 1.14i)T \)
good7 \( 1 + (1.44 - 2.50i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.395 - 0.684i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.04 - 2.33i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.89T + 17T^{2} \)
19 \( 1 - 4.55iT - 19T^{2} \)
23 \( 1 + (-1.15 + 0.666i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.29 - 1.32i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.26 - 1.30i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.44iT - 37T^{2} \)
41 \( 1 + (5.91 - 3.41i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.95 + 10.3i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.60 + 1.50i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.70T + 53T^{2} \)
59 \( 1 + (5.29 + 9.17i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.869 - 1.50i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.76 - 6.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.88T + 71T^{2} \)
73 \( 1 + 6.50iT - 73T^{2} \)
79 \( 1 + (-3.30 - 1.91i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.33 + 4.81i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.00741iT - 89T^{2} \)
97 \( 1 + (-5.74 - 3.31i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.78458915035994690554443379464, −12.15920969935695636383737669496, −11.82618281448055699366091202983, −10.04812936775454731099806323345, −9.063820474814386001399984640323, −7.83145620849515372673433134495, −6.77354069694340218896370248709, −5.57004215104239280399113000698, −4.79730319625462137688199340951, −3.48894374046698291371105393004, 0.53799401805692379535198448989, 3.00939115007134815258805072065, 4.32201000949095614549058112478, 5.50898735715800976868786597578, 6.85972205457627521202772273319, 7.71858414684259818189497097780, 9.726804677216213560077668569222, 10.51783411560715392077080342644, 11.21240057720464598724556745503, 12.19868499830212668489444870237

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