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Properties

Label 2-180-20.19-c2-0-27
Degree $2$
Conductor $180$
Sign $-0.769 - 0.639i$
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 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (−4.69 + 1.73i)5-s − 9.38·7-s − 7.99·8-s + (−1.69 + 9.85i)10-s + 16.2i·11-s − 16.2i·13-s + (−9.38 + 16.2i)14-s + (−8 + 13.8i)16-s − 10.3i·17-s − 20.7i·19-s + (15.3 + 12.7i)20-s + (28.1 + 16.2i)22-s − 28·23-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.938 + 0.346i)5-s − 1.34·7-s − 0.999·8-s + (−0.169 + 0.985i)10-s + 1.47i·11-s − 1.24i·13-s + (−0.670 + 1.16i)14-s + (−0.5 + 0.866i)16-s − 0.611i·17-s − 1.09i·19-s + (0.769 + 0.639i)20-s + (1.27 + 0.738i)22-s − 1.21·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 - 0.639i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.769 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.769 - 0.639i$
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: \((2,\ 180,\ (\ :1),\ -0.769 - 0.639i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.100105 + 0.277050i\)
\(L(\frac12)\) \(\approx\) \(0.100105 + 0.277050i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 \)
5 \( 1 + (4.69 - 1.73i)T \)
good7 \( 1 + 9.38T + 49T^{2} \)
11 \( 1 - 16.2iT - 121T^{2} \)
13 \( 1 + 16.2iT - 169T^{2} \)
17 \( 1 + 10.3iT - 289T^{2} \)
19 \( 1 + 20.7iT - 361T^{2} \)
23 \( 1 + 28T + 529T^{2} \)
29 \( 1 + 9.38T + 841T^{2} \)
31 \( 1 - 34.6iT - 961T^{2} \)
37 \( 1 + 48.7iT - 1.36e3T^{2} \)
41 \( 1 - 18.7T + 1.68e3T^{2} \)
43 \( 1 + 37.5T + 1.84e3T^{2} \)
47 \( 1 + 4T + 2.20e3T^{2} \)
53 \( 1 - 31.1iT - 2.80e3T^{2} \)
59 \( 1 + 16.2iT - 3.48e3T^{2} \)
61 \( 1 + 58T + 3.72e3T^{2} \)
67 \( 1 - 18.7T + 4.48e3T^{2} \)
71 \( 1 + 97.4iT - 5.04e3T^{2} \)
73 \( 1 - 5.32e3T^{2} \)
79 \( 1 - 6.92iT - 6.24e3T^{2} \)
83 \( 1 - 32T + 6.88e3T^{2} \)
89 \( 1 - 75.0T + 7.92e3T^{2} \)
97 \( 1 - 162. iT - 9.40e3T^{2} \)
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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

−12.12087017653052778804661382677, −10.83578486218282042107474172253, −10.04971697415900601213610607695, −9.160227629097614223852503432467, −7.55354549826721477104434466304, −6.47551655124114185345839635490, −4.96997224866132645759117577788, −3.71737038521163428511182725934, −2.67004404795703394759599532448, −0.14331428505378910520377539383, 3.39179343474079249387730381640, 4.13849125708395487714501607999, 5.85351783063654029674766736161, 6.58604519227008378333442878138, 7.912102897513532158499846574322, 8.691295848838666050618818482546, 9.791001491669682409268069282912, 11.43745738486121253178927960975, 12.19998599351468460791781126503, 13.15197126999979327231844952107

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