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Properties

Label 2-180-20.19-c2-0-25
Degree $2$
Conductor $180$
Sign $-0.648 + 0.761i$
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  + (1.52 − 1.29i)2-s + (0.637 − 3.94i)4-s + (−4.27 − 2.59i)5-s + 0.837·7-s + (−4.14 − 6.83i)8-s + (−9.87 + 1.59i)10-s − 15.7i·11-s − 5.18i·13-s + (1.27 − 1.08i)14-s + (−15.1 − 5.03i)16-s + 27.3i·17-s − 17.9i·19-s + (−12.9 + 15.2i)20-s + (−20.4 − 24.0i)22-s + 19.1·23-s + ⋯
L(s)  = 1  + (0.761 − 0.648i)2-s + (0.159 − 0.987i)4-s + (−0.854 − 0.518i)5-s + 0.119·7-s + (−0.518 − 0.854i)8-s + (−0.987 + 0.159i)10-s − 1.43i·11-s − 0.398i·13-s + (0.0910 − 0.0775i)14-s + (−0.949 − 0.314i)16-s + 1.60i·17-s − 0.945i·19-s + (−0.648 + 0.761i)20-s + (−0.930 − 1.09i)22-s + 0.830·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 + 0.761i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.648 + 0.761i$
Analytic conductor: \(4.90464\)
Root analytic conductor: \(2.21464\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1),\ -0.648 + 0.761i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.748172 - 1.61964i\)
\(L(\frac12)\) \(\approx\) \(0.748172 - 1.61964i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.52 + 1.29i)T \)
3 \( 1 \)
5 \( 1 + (4.27 + 2.59i)T \)
good7 \( 1 - 0.837T + 49T^{2} \)
11 \( 1 + 15.7iT - 121T^{2} \)
13 \( 1 + 5.18iT - 169T^{2} \)
17 \( 1 - 27.3iT - 289T^{2} \)
19 \( 1 + 17.9iT - 361T^{2} \)
23 \( 1 - 19.1T + 529T^{2} \)
29 \( 1 - 45.6T + 841T^{2} \)
31 \( 1 + 13.6iT - 961T^{2} \)
37 \( 1 - 15.5iT - 1.36e3T^{2} \)
41 \( 1 + 13.2T + 1.68e3T^{2} \)
43 \( 1 - 27.9T + 1.84e3T^{2} \)
47 \( 1 - 55.6T + 2.20e3T^{2} \)
53 \( 1 + 15.5iT - 2.80e3T^{2} \)
59 \( 1 - 87.6iT - 3.48e3T^{2} \)
61 \( 1 - 38T + 3.72e3T^{2} \)
67 \( 1 - 92.2T + 4.48e3T^{2} \)
71 \( 1 + 130. iT - 5.04e3T^{2} \)
73 \( 1 + 54.7iT - 5.32e3T^{2} \)
79 \( 1 - 13.6iT - 6.24e3T^{2} \)
83 \( 1 + 59.0T + 6.88e3T^{2} \)
89 \( 1 + 39.8T + 7.92e3T^{2} \)
97 \( 1 - 168. iT - 9.40e3T^{2} \)
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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

−12.11234643926914395499908307533, −11.16805011745389464187734154297, −10.54864158624909902841076186675, −9.001558057102161683889985198825, −8.139263152582364598090650874327, −6.53106182403529467264691414485, −5.35443821781891416711532013046, −4.16880497506865574118289681564, −3.04249963774564307185595012233, −0.885931110136733617412145854802, 2.72500804699581193060984271347, 4.15018697163166027761421982132, 5.08465120676890535342822308687, 6.77748269608075191580155358129, 7.29480656905864131084920677238, 8.387708133826134010688676077935, 9.757138917286647718668356099017, 11.15398259791927374100888228759, 12.03646369982062248838871883104, 12.65093529329051323846269584666

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