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Properties

Label 2-180-4.3-c2-0-15
Degree $2$
Conductor $180$
Sign $-0.809 + 0.587i$
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  + (−0.618 − 1.90i)2-s + (−3.23 + 2.35i)4-s + 2.23·5-s − 5.25i·7-s + (6.47 + 4.70i)8-s + (−1.38 − 4.25i)10-s − 19.9i·11-s − 8.47·13-s + (−9.99 + 3.24i)14-s + (4.94 − 15.2i)16-s − 11.8·17-s − 15.2i·19-s + (−7.23 + 5.25i)20-s + (−37.8 + 12.3i)22-s + 0.555i·23-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + 0.447·5-s − 0.751i·7-s + (0.809 + 0.587i)8-s + (−0.138 − 0.425i)10-s − 1.81i·11-s − 0.651·13-s + (−0.714 + 0.232i)14-s + (0.309 − 0.951i)16-s − 0.699·17-s − 0.800i·19-s + (−0.361 + 0.262i)20-s + (−1.72 + 0.559i)22-s + 0.0241i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.809 + 0.587i$
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: \((2,\ 180,\ (\ :1),\ -0.809 + 0.587i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.320078 - 0.985099i\)
\(L(\frac12)\) \(\approx\) \(0.320078 - 0.985099i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.618 + 1.90i)T \)
3 \( 1 \)
5 \( 1 - 2.23T \)
good7 \( 1 + 5.25iT - 49T^{2} \)
11 \( 1 + 19.9iT - 121T^{2} \)
13 \( 1 + 8.47T + 169T^{2} \)
17 \( 1 + 11.8T + 289T^{2} \)
19 \( 1 + 15.2iT - 361T^{2} \)
23 \( 1 - 0.555iT - 529T^{2} \)
29 \( 1 - 10.9T + 841T^{2} \)
31 \( 1 + 8.29iT - 961T^{2} \)
37 \( 1 + 18.3T + 1.36e3T^{2} \)
41 \( 1 - 14.5T + 1.68e3T^{2} \)
43 \( 1 - 22.2iT - 1.84e3T^{2} \)
47 \( 1 + 53.3iT - 2.20e3T^{2} \)
53 \( 1 - 66.3T + 2.80e3T^{2} \)
59 \( 1 + 17.4iT - 3.48e3T^{2} \)
61 \( 1 - 90.1T + 3.72e3T^{2} \)
67 \( 1 - 50.2iT - 4.48e3T^{2} \)
71 \( 1 - 80.7iT - 5.04e3T^{2} \)
73 \( 1 + 5.55T + 5.32e3T^{2} \)
79 \( 1 + 13.8iT - 6.24e3T^{2} \)
83 \( 1 - 76.2iT - 6.88e3T^{2} \)
89 \( 1 - 111.T + 7.92e3T^{2} \)
97 \( 1 + 92.8T + 9.40e3T^{2} \)
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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

−11.78955070310046609880093458285, −10.98955254096955807933798083685, −10.22810824860351318072284519262, −9.112993583063023573389939760877, −8.269546099363164904621286182288, −6.90661049569924219153260645798, −5.34205980062353352062302120108, −3.94213392995068983698074401931, −2.60865586196564007459598000088, −0.69676874048552710604398229926, 2.05194167619393412273473921435, 4.45061087105230734621054161227, 5.45372050141977323325365379743, 6.65144728692677667916969098929, 7.56476722132588954222966461423, 8.810136251039589896746596715265, 9.657470803896341901102879604985, 10.41480574442700365508831385348, 12.11520730162970866768238867634, 12.83675816336909171465557639654

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