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Properties

Label 2-180-4.3-c8-0-40
Degree $2$
Conductor $180$
Sign $0.577 + 0.816i$
Analytic cond. $73.3281$
Root an. cond. $8.56318$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.35 − 14.2i)2-s + (−147. − 208. i)4-s + 279.·5-s + 3.54e3i·7-s + (−4.05e3 + 567. i)8-s + (2.05e3 − 3.97e3i)10-s − 1.59e4i·11-s − 2.41e4·13-s + (5.03e4 + 2.60e4i)14-s + (−2.17e4 + 6.18e4i)16-s − 4.38e4·17-s + 5.09e4i·19-s + (−4.13e4 − 5.83e4i)20-s + (−2.26e5 − 1.17e5i)22-s − 2.70e5i·23-s + ⋯
L(s)  = 1  + (0.459 − 0.888i)2-s + (−0.577 − 0.816i)4-s + 0.447·5-s + 1.47i·7-s + (−0.990 + 0.138i)8-s + (0.205 − 0.397i)10-s − 1.09i·11-s − 0.846·13-s + (1.30 + 0.677i)14-s + (−0.332 + 0.943i)16-s − 0.524·17-s + 0.390i·19-s + (−0.258 − 0.364i)20-s + (−0.968 − 0.501i)22-s − 0.967i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(73.3281\)
Root analytic conductor: \(8.56318\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :4),\ 0.577 + 0.816i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.484376585\)
\(L(\frac12)\) \(\approx\) \(2.484376585\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.35 + 14.2i)T \)
3 \( 1 \)
5 \( 1 - 279.T \)
good7 \( 1 - 3.54e3iT - 5.76e6T^{2} \)
11 \( 1 + 1.59e4iT - 2.14e8T^{2} \)
13 \( 1 + 2.41e4T + 8.15e8T^{2} \)
17 \( 1 + 4.38e4T + 6.97e9T^{2} \)
19 \( 1 - 5.09e4iT - 1.69e10T^{2} \)
23 \( 1 + 2.70e5iT - 7.83e10T^{2} \)
29 \( 1 - 1.32e6T + 5.00e11T^{2} \)
31 \( 1 - 1.18e6iT - 8.52e11T^{2} \)
37 \( 1 - 2.97e6T + 3.51e12T^{2} \)
41 \( 1 - 4.92e6T + 7.98e12T^{2} \)
43 \( 1 + 2.86e6iT - 1.16e13T^{2} \)
47 \( 1 - 1.26e4iT - 2.38e13T^{2} \)
53 \( 1 - 5.50e6T + 6.22e13T^{2} \)
59 \( 1 - 6.68e6iT - 1.46e14T^{2} \)
61 \( 1 + 2.50e6T + 1.91e14T^{2} \)
67 \( 1 - 3.86e6iT - 4.06e14T^{2} \)
71 \( 1 + 3.21e7iT - 6.45e14T^{2} \)
73 \( 1 - 1.91e7T + 8.06e14T^{2} \)
79 \( 1 - 5.33e7iT - 1.51e15T^{2} \)
83 \( 1 + 6.22e6iT - 2.25e15T^{2} \)
89 \( 1 + 3.83e7T + 3.93e15T^{2} \)
97 \( 1 - 7.87e7T + 7.83e15T^{2} \)
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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

−11.12497397264902521878231023527, −10.15907464974909364481571360320, −9.093918361698729718428637504479, −8.439085415472726007070301548539, −6.38468079066569406907212802341, −5.60331202322360582553224840111, −4.56933178039304979812113246694, −2.91016567459353602034555784831, −2.30152321538997838838588124718, −0.78146115723337088063762512450, 0.72984626349804801268315447527, 2.55984629109237760186142805124, 4.16569406134087982741645422761, 4.78146591753072595665450769413, 6.24432984906721052043565763156, 7.20502125119750996572855274150, 7.81664348277848271509608065119, 9.382506251770112345401456046402, 10.06662121136695932769985174135, 11.42326366441551524684019982626

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