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Properties

Label 2-180-4.3-c6-0-12
Degree $2$
Conductor $180$
Sign $-0.941 + 0.336i$
Analytic cond. $41.4097$
Root an. cond. $6.43503$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 7.88i)2-s + (−60.2 + 21.5i)4-s + 55.9·5-s + 86.8i·7-s + (−251. − 445. i)8-s + (76.2 + 440. i)10-s + 1.54e3i·11-s + 3.18e3·13-s + (−685. + 118. i)14-s + (3.17e3 − 2.59e3i)16-s − 4.06e3·17-s + 536. i·19-s + (−3.36e3 + 1.20e3i)20-s + (−1.21e4 + 2.10e3i)22-s + 1.81e4i·23-s + ⋯
L(s)  = 1  + (0.170 + 0.985i)2-s + (−0.941 + 0.336i)4-s + 0.447·5-s + 0.253i·7-s + (−0.491 − 0.870i)8-s + (0.0762 + 0.440i)10-s + 1.16i·11-s + 1.45·13-s + (−0.249 + 0.0431i)14-s + (0.774 − 0.632i)16-s − 0.826·17-s + 0.0782i·19-s + (−0.421 + 0.150i)20-s + (−1.14 + 0.198i)22-s + 1.48i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.941 + 0.336i$
Analytic conductor: \(41.4097\)
Root analytic conductor: \(6.43503\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :3),\ -0.941 + 0.336i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.341847735\)
\(L(\frac12)\) \(\approx\) \(1.341847735\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 7.88i)T \)
3 \( 1 \)
5 \( 1 - 55.9T \)
good7 \( 1 - 86.8iT - 1.17e5T^{2} \)
11 \( 1 - 1.54e3iT - 1.77e6T^{2} \)
13 \( 1 - 3.18e3T + 4.82e6T^{2} \)
17 \( 1 + 4.06e3T + 2.41e7T^{2} \)
19 \( 1 - 536. iT - 4.70e7T^{2} \)
23 \( 1 - 1.81e4iT - 1.48e8T^{2} \)
29 \( 1 + 1.49e4T + 5.94e8T^{2} \)
31 \( 1 + 1.17e4iT - 8.87e8T^{2} \)
37 \( 1 + 3.49e4T + 2.56e9T^{2} \)
41 \( 1 + 1.08e5T + 4.75e9T^{2} \)
43 \( 1 - 9.19e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.72e4iT - 1.07e10T^{2} \)
53 \( 1 + 2.38e5T + 2.21e10T^{2} \)
59 \( 1 + 9.56e4iT - 4.21e10T^{2} \)
61 \( 1 + 1.44e5T + 5.15e10T^{2} \)
67 \( 1 + 4.94e5iT - 9.04e10T^{2} \)
71 \( 1 + 2.63e5iT - 1.28e11T^{2} \)
73 \( 1 + 2.25e5T + 1.51e11T^{2} \)
79 \( 1 - 3.18e5iT - 2.43e11T^{2} \)
83 \( 1 - 8.80e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.98e5T + 4.96e11T^{2} \)
97 \( 1 + 1.41e5T + 8.32e11T^{2} \)
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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

−12.39985821185024426763206071881, −11.12369293990381662032813230721, −9.765976003819721153000083609803, −9.025277954626590433529018376994, −7.930316792911069832940015266072, −6.81233681442997307367305424906, −5.90057173609208396027360250490, −4.79571682078182778008355592470, −3.53795310968261484563677541961, −1.62673495682127973953475892758, 0.35942579627371075907471547334, 1.61168102400419741609718894899, 3.04320460309818636471053601093, 4.13365724676679321435553370488, 5.51273925664482030132476234481, 6.53327228083213209436508901406, 8.481967618646026300723799678552, 8.929270996322029812435477833847, 10.40663338177993577824842011227, 10.88553550436335711387347566416

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