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Properties

Label 2-180-4.3-c2-0-16
Degree $2$
Conductor $180$
Sign $0.985 + 0.168i$
Analytic cond. $4.90464$
Root an. cond. $2.21464$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.99 + 0.169i)2-s + (3.94 + 0.675i)4-s + 2.23·5-s − 12.3i·7-s + (7.74 + 2.01i)8-s + (4.45 + 0.378i)10-s + 11.0i·11-s + 2.82·13-s + (2.10 − 24.7i)14-s + (15.0 + 5.32i)16-s − 6.52·17-s + 27.9i·19-s + (8.81 + 1.51i)20-s + (−1.87 + 22.0i)22-s − 7.90i·23-s + ⋯
L(s)  = 1  + (0.996 + 0.0847i)2-s + (0.985 + 0.168i)4-s + 0.447·5-s − 1.77i·7-s + (0.967 + 0.251i)8-s + (0.445 + 0.0378i)10-s + 1.00i·11-s + 0.216·13-s + (0.150 − 1.76i)14-s + (0.942 + 0.332i)16-s − 0.383·17-s + 1.47i·19-s + (0.440 + 0.0755i)20-s + (−0.0850 + 1.00i)22-s − 0.343i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.985 + 0.168i$
Analytic conductor: \(4.90464\)
Root analytic conductor: \(2.21464\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1),\ 0.985 + 0.168i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.86785 - 0.243853i\)
\(L(\frac12)\) \(\approx\) \(2.86785 - 0.243853i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 - 0.169i)T \)
3 \( 1 \)
5 \( 1 - 2.23T \)
good7 \( 1 + 12.3iT - 49T^{2} \)
11 \( 1 - 11.0iT - 121T^{2} \)
13 \( 1 - 2.82T + 169T^{2} \)
17 \( 1 + 6.52T + 289T^{2} \)
19 \( 1 - 27.9iT - 361T^{2} \)
23 \( 1 + 7.90iT - 529T^{2} \)
29 \( 1 + 50.7T + 841T^{2} \)
31 \( 1 + 36.3iT - 961T^{2} \)
37 \( 1 + 18.9T + 1.36e3T^{2} \)
41 \( 1 + 5.30T + 1.68e3T^{2} \)
43 \( 1 - 45.5iT - 1.84e3T^{2} \)
47 \( 1 - 11.7iT - 2.20e3T^{2} \)
53 \( 1 + 41.1T + 2.80e3T^{2} \)
59 \( 1 - 10.7iT - 3.48e3T^{2} \)
61 \( 1 - 56.1T + 3.72e3T^{2} \)
67 \( 1 - 16.1iT - 4.48e3T^{2} \)
71 \( 1 - 66.1iT - 5.04e3T^{2} \)
73 \( 1 - 15.6T + 5.32e3T^{2} \)
79 \( 1 + 123. iT - 6.24e3T^{2} \)
83 \( 1 + 99.6iT - 6.88e3T^{2} \)
89 \( 1 + 101.T + 7.92e3T^{2} \)
97 \( 1 - 127.T + 9.40e3T^{2} \)
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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

−12.78166959502743232009401051796, −11.46574148973780660529537933067, −10.52447423821151611521272219648, −9.780117742623094533072701784284, −7.83793333994463450474741257062, −7.09462002922511308773231349895, −5.99361825325606868085044658429, −4.55997107543300771463601370119, −3.69293612372741852465916949608, −1.75094143280538136637803540220, 2.12631096288277793706635003876, 3.28514627631965387733431727639, 5.12639584927953566803378220584, 5.78637033860650186731953295504, 6.85160814201763112340512736474, 8.508169303414688359560981129277, 9.362276719488481858862116402750, 10.91590562499004635896610585918, 11.56211579107659668118345717832, 12.56956566938289769138265301436

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