Properties

Label 2-180-36.11-c1-0-4
Degree $2$
Conductor $180$
Sign $-0.533 - 0.846i$
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  + (−0.962 + 1.03i)2-s + (0.968 + 1.43i)3-s + (−0.147 − 1.99i)4-s + (−0.866 + 0.5i)5-s + (−2.42 − 0.378i)6-s + (1.90 + 1.09i)7-s + (2.20 + 1.76i)8-s + (−1.12 + 2.78i)9-s + (0.315 − 1.37i)10-s + (−2.41 + 4.17i)11-s + (2.72 − 2.14i)12-s + (0.364 + 0.630i)13-s + (−2.96 + 0.913i)14-s + (−1.55 − 0.759i)15-s + (−3.95 + 0.588i)16-s − 6.46i·17-s + ⋯
L(s)  = 1  + (−0.680 + 0.732i)2-s + (0.559 + 0.829i)3-s + (−0.0737 − 0.997i)4-s + (−0.387 + 0.223i)5-s + (−0.987 − 0.154i)6-s + (0.718 + 0.414i)7-s + (0.780 + 0.624i)8-s + (−0.374 + 0.927i)9-s + (0.0997 − 0.435i)10-s + (−0.727 + 1.25i)11-s + (0.785 − 0.618i)12-s + (0.101 + 0.174i)13-s + (−0.793 + 0.244i)14-s + (−0.401 − 0.196i)15-s + (−0.989 + 0.147i)16-s − 1.56i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.533 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.533 - 0.846i$
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: \((2,\ 180,\ (\ :1/2),\ -0.533 - 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.450922 + 0.817095i\)
\(L(\frac12)\) \(\approx\) \(0.450922 + 0.817095i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.962 - 1.03i)T \)
3 \( 1 + (-0.968 - 1.43i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
good7 \( 1 + (-1.90 - 1.09i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.41 - 4.17i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.364 - 0.630i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.46iT - 17T^{2} \)
19 \( 1 + 3.21iT - 19T^{2} \)
23 \( 1 + (-3.67 - 6.36i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.238 + 0.137i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.796 + 0.459i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.84T + 37T^{2} \)
41 \( 1 + (-8.66 + 5.00i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.45 - 0.842i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.39 + 5.88i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.913iT - 53T^{2} \)
59 \( 1 + (5.14 + 8.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.27 - 7.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.49 - 0.862i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.71T + 71T^{2} \)
73 \( 1 - 8.12T + 73T^{2} \)
79 \( 1 + (-1.83 - 1.06i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.54 + 6.14i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 1.54iT - 89T^{2} \)
97 \( 1 + (3.07 - 5.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.34696924085502503830210968266, −11.59903187717291532276110563731, −10.84248653912672053070709071918, −9.659347753830245955058822352505, −9.088350876736802836668656598702, −7.82597150688500162995309342946, −7.23768411411589936958734502517, −5.32842512243029178411321698048, −4.58191628900188232161358258003, −2.45960619105250757598561743098, 1.11275126296457977439677035289, 2.81579197662576615436632661014, 4.14654137221412086500009969577, 6.19767480909862597609911379963, 7.85613482078996397860552533077, 8.079979429996189817824540072067, 9.058309228839898951714808735591, 10.61613060517013456701849987808, 11.16997376457548307055917264489, 12.44409283874906334561739888763

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