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Properties

Label 2-180-180.59-c1-0-3
Degree $2$
Conductor $180$
Sign $0.997 - 0.0641i$
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  + (−1.24 − 0.668i)2-s + (−1.66 + 0.461i)3-s + (1.10 + 1.66i)4-s + (−1.69 − 1.46i)5-s + (2.38 + 0.540i)6-s + (0.667 + 1.15i)7-s + (−0.265 − 2.81i)8-s + (2.57 − 1.54i)9-s + (1.13 + 2.95i)10-s + (2.18 + 3.78i)11-s + (−2.61 − 2.27i)12-s + (3.56 + 2.05i)13-s + (−0.0591 − 1.88i)14-s + (3.49 + 1.65i)15-s + (−1.55 + 3.68i)16-s + 6.45·17-s + ⋯
L(s)  = 1  + (−0.881 − 0.472i)2-s + (−0.963 + 0.266i)3-s + (0.553 + 0.832i)4-s + (−0.756 − 0.653i)5-s + (0.975 + 0.220i)6-s + (0.252 + 0.436i)7-s + (−0.0939 − 0.995i)8-s + (0.858 − 0.513i)9-s + (0.358 + 0.933i)10-s + (0.659 + 1.14i)11-s + (−0.755 − 0.655i)12-s + (0.987 + 0.570i)13-s + (−0.0158 − 0.504i)14-s + (0.903 + 0.428i)15-s + (−0.387 + 0.921i)16-s + 1.56·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.997 - 0.0641i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.997 - 0.0641i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.573008 + 0.0184074i\)
\(L(\frac12)\) \(\approx\) \(0.573008 + 0.0184074i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.24 + 0.668i)T \)
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

−12.10674581720289207666939262260, −11.76445645307476994822002561767, −10.83080528639297059062110702501, −9.632394026180610026969114589234, −8.886410879777022063499559817500, −7.62090145459835551015348013263, −6.58533408320372404626865747852, −4.95037718406245054191431640963, −3.75392180158536274482221515844, −1.29598216734855615121422637421, 1.01136589964481395747910910488, 3.70509699302834885449838123773, 5.67647538065835104141360779469, 6.36056303189433309368976732514, 7.65490419425035539666444348937, 8.167392390812806040783200702939, 9.869757850105342452374443425809, 10.79384054769607756432243562518, 11.33294296357292929007081251167, 12.25159190394394505899326904531

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