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Properties

Label 2-180-4.3-c2-0-9
Degree $2$
Conductor $180$
Sign $0.757 + 0.652i$
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  + (−1.87 − 0.696i)2-s + (3.02 + 2.61i)4-s + 2.23·5-s − 5.46i·7-s + (−3.86 − 7.00i)8-s + (−4.19 − 1.55i)10-s + 11.0i·11-s + 10.1·13-s + (−3.80 + 10.2i)14-s + (2.35 + 15.8i)16-s + 24.4·17-s − 23.7i·19-s + (6.77 + 5.84i)20-s + (7.69 − 20.6i)22-s − 37.2i·23-s + ⋯
L(s)  = 1  + (−0.937 − 0.348i)2-s + (0.757 + 0.652i)4-s + 0.447·5-s − 0.781i·7-s + (−0.482 − 0.875i)8-s + (−0.419 − 0.155i)10-s + 1.00i·11-s + 0.778·13-s + (−0.272 + 0.732i)14-s + (0.147 + 0.989i)16-s + 1.43·17-s − 1.25i·19-s + (0.338 + 0.292i)20-s + (0.349 − 0.940i)22-s − 1.61i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.757 + 0.652i$
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: \((2,\ 180,\ (\ :1),\ 0.757 + 0.652i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.03904 - 0.386061i\)
\(L(\frac12)\) \(\approx\) \(1.03904 - 0.386061i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.87 + 0.696i)T \)
3 \( 1 \)
5 \( 1 - 2.23T \)
good7 \( 1 + 5.46iT - 49T^{2} \)
11 \( 1 - 11.0iT - 121T^{2} \)
13 \( 1 - 10.1T + 169T^{2} \)
17 \( 1 - 24.4T + 289T^{2} \)
19 \( 1 + 23.7iT - 361T^{2} \)
23 \( 1 + 37.2iT - 529T^{2} \)
29 \( 1 - 25.7T + 841T^{2} \)
31 \( 1 - 4.83iT - 961T^{2} \)
37 \( 1 - 35.6T + 1.36e3T^{2} \)
41 \( 1 - 9.30T + 1.68e3T^{2} \)
43 \( 1 - 70.0iT - 1.84e3T^{2} \)
47 \( 1 + 38.0iT - 2.20e3T^{2} \)
53 \( 1 + 55.7T + 2.80e3T^{2} \)
59 \( 1 - 55.5iT - 3.48e3T^{2} \)
61 \( 1 + 82.2T + 3.72e3T^{2} \)
67 \( 1 + 104. iT - 4.48e3T^{2} \)
71 \( 1 - 76.7iT - 5.04e3T^{2} \)
73 \( 1 + 93.5T + 5.32e3T^{2} \)
79 \( 1 - 49.3iT - 6.24e3T^{2} \)
83 \( 1 - 72.3iT - 6.88e3T^{2} \)
89 \( 1 + 115.T + 7.92e3T^{2} \)
97 \( 1 + 72.9T + 9.40e3T^{2} \)
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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

−12.24345285523119542503517410104, −11.03525588226032184246859734296, −10.26078776408249232870047905492, −9.482869336087374536805704799789, −8.323905537357410260432059548786, −7.27554758377647133264799077501, −6.31320241278321208628404069312, −4.45020735844203609799023981616, −2.81536793444476321906241511140, −1.09960482851967168882947662812, 1.40494839326151203836650290626, 3.18612944744639846952569206082, 5.65114988786692094302632187678, 6.01389540750696882848330441433, 7.64582316245026637846067310131, 8.506578683626581191537672257180, 9.447484463713484472738835716565, 10.35058720774684138564735040850, 11.40825646612759371263535241233, 12.27856509498182514451860040732

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