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Properties

Label 2-180-60.47-c2-0-9
Degree $2$
Conductor $180$
Sign $0.338 - 0.940i$
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  + (1.97 + 0.321i)2-s + (3.79 + 1.26i)4-s + (0.196 + 4.99i)5-s + (−8.42 + 8.42i)7-s + (7.08 + 3.72i)8-s + (−1.21 + 9.92i)10-s + 0.926·11-s + (8.10 − 8.10i)13-s + (−19.3 + 13.9i)14-s + (12.7 + 9.62i)16-s + (20.7 − 20.7i)17-s − 13.8·19-s + (−5.59 + 19.2i)20-s + (1.82 + 0.297i)22-s + (22.9 − 22.9i)23-s + ⋯
L(s)  = 1  + (0.987 + 0.160i)2-s + (0.948 + 0.317i)4-s + (0.0393 + 0.999i)5-s + (−1.20 + 1.20i)7-s + (0.885 + 0.465i)8-s + (−0.121 + 0.992i)10-s + 0.0841·11-s + (0.623 − 0.623i)13-s + (−1.38 + 0.994i)14-s + (0.798 + 0.601i)16-s + (1.21 − 1.21i)17-s − 0.728·19-s + (−0.279 + 0.960i)20-s + (0.0830 + 0.0135i)22-s + (0.999 − 0.999i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 - 0.940i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.338 - 0.940i$
Analytic conductor: \(4.90464\)
Root analytic conductor: \(2.21464\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1),\ 0.338 - 0.940i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.05557 + 1.44506i\)
\(L(\frac12)\) \(\approx\) \(2.05557 + 1.44506i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.97 - 0.321i)T \)
3 \( 1 \)
5 \( 1 + (-0.196 - 4.99i)T \)
good7 \( 1 + (8.42 - 8.42i)T - 49iT^{2} \)
11 \( 1 - 0.926T + 121T^{2} \)
13 \( 1 + (-8.10 + 8.10i)T - 169iT^{2} \)
17 \( 1 + (-20.7 + 20.7i)T - 289iT^{2} \)
19 \( 1 + 13.8T + 361T^{2} \)
23 \( 1 + (-22.9 + 22.9i)T - 529iT^{2} \)
29 \( 1 - 12.0T + 841T^{2} \)
31 \( 1 - 27.6iT - 961T^{2} \)
37 \( 1 + (-14.0 - 14.0i)T + 1.36e3iT^{2} \)
41 \( 1 - 10.4iT - 1.68e3T^{2} \)
43 \( 1 + (44.6 + 44.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (8.50 + 8.50i)T + 2.20e3iT^{2} \)
53 \( 1 + (7.70 + 7.70i)T + 2.80e3iT^{2} \)
59 \( 1 - 12.6iT - 3.48e3T^{2} \)
61 \( 1 + 13.7T + 3.72e3T^{2} \)
67 \( 1 + (-75.3 + 75.3i)T - 4.48e3iT^{2} \)
71 \( 1 + 13.5T + 5.04e3T^{2} \)
73 \( 1 + (43.6 - 43.6i)T - 5.32e3iT^{2} \)
79 \( 1 - 71.8T + 6.24e3T^{2} \)
83 \( 1 + (21.6 - 21.6i)T - 6.88e3iT^{2} \)
89 \( 1 + 98.6T + 7.92e3T^{2} \)
97 \( 1 + (27.6 + 27.6i)T + 9.40e3iT^{2} \)
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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

−12.62150993695955965134014942311, −11.91512149591311785392135354701, −10.80791584523132748900010537822, −9.837477757679865334745240322437, −8.423751241061476255953665259901, −6.97610598273763999279740339846, −6.26492924317235747883358040843, −5.25592948810283913383735222094, −3.35219707055608948171136186004, −2.69892354521630943315189029738, 1.27230976356102384405372460237, 3.49682128438499588379062787015, 4.30201166560570249179276985096, 5.77209815290378162704273707937, 6.70955832974120708705070813793, 7.945263257241490827093803455140, 9.473721488731506901369571263489, 10.34605713940288384891387334830, 11.42707084797511519454788416479, 12.63043419369383067369930684028

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