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Properties

Label 2-180-4.3-c8-0-76
Degree $2$
Conductor $180$
Sign $-0.749 + 0.661i$
Analytic cond. $73.3281$
Root an. cond. $8.56318$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (14.9 − 5.65i)2-s + (191. − 169. i)4-s + 279.·5-s − 2.97e3i·7-s + (1.91e3 − 3.62e3i)8-s + (4.18e3 − 1.58e3i)10-s − 2.80e4i·11-s + 3.56e4·13-s + (−1.68e4 − 4.45e4i)14-s + (8.16e3 − 6.50e4i)16-s − 5.93e4·17-s + 1.46e5i·19-s + (5.36e4 − 4.73e4i)20-s + (−1.58e5 − 4.20e5i)22-s + 3.09e5i·23-s + ⋯
L(s)  = 1  + (0.935 − 0.353i)2-s + (0.749 − 0.661i)4-s + 0.447·5-s − 1.23i·7-s + (0.467 − 0.884i)8-s + (0.418 − 0.158i)10-s − 1.91i·11-s + 1.24·13-s + (−0.438 − 1.15i)14-s + (0.124 − 0.992i)16-s − 0.710·17-s + 1.12i·19-s + (0.335 − 0.295i)20-s + (−0.677 − 1.79i)22-s + 1.10i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.749 + 0.661i$
Analytic conductor: \(73.3281\)
Root analytic conductor: \(8.56318\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :4),\ -0.749 + 0.661i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(4.172938380\)
\(L(\frac12)\) \(\approx\) \(4.172938380\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-14.9 + 5.65i)T \)
3 \( 1 \)
5 \( 1 - 279.T \)
good7 \( 1 + 2.97e3iT - 5.76e6T^{2} \)
11 \( 1 + 2.80e4iT - 2.14e8T^{2} \)
13 \( 1 - 3.56e4T + 8.15e8T^{2} \)
17 \( 1 + 5.93e4T + 6.97e9T^{2} \)
19 \( 1 - 1.46e5iT - 1.69e10T^{2} \)
23 \( 1 - 3.09e5iT - 7.83e10T^{2} \)
29 \( 1 + 1.02e6T + 5.00e11T^{2} \)
31 \( 1 + 1.62e5iT - 8.52e11T^{2} \)
37 \( 1 - 8.64e5T + 3.51e12T^{2} \)
41 \( 1 + 1.51e6T + 7.98e12T^{2} \)
43 \( 1 + 2.39e6iT - 1.16e13T^{2} \)
47 \( 1 + 2.15e6iT - 2.38e13T^{2} \)
53 \( 1 - 9.78e6T + 6.22e13T^{2} \)
59 \( 1 - 7.13e6iT - 1.46e14T^{2} \)
61 \( 1 + 5.43e6T + 1.91e14T^{2} \)
67 \( 1 - 1.02e7iT - 4.06e14T^{2} \)
71 \( 1 + 3.55e7iT - 6.45e14T^{2} \)
73 \( 1 + 3.90e6T + 8.06e14T^{2} \)
79 \( 1 + 3.17e6iT - 1.51e15T^{2} \)
83 \( 1 + 4.52e7iT - 2.25e15T^{2} \)
89 \( 1 + 4.70e7T + 3.93e15T^{2} \)
97 \( 1 + 1.22e7T + 7.83e15T^{2} \)
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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

−10.95708568621917549189002115426, −10.21391402037601807930061744684, −8.824143595949129650481085827114, −7.48858066342319500241417128975, −6.22054388928518242121867458231, −5.58920869251927987723537724535, −3.93513767171014666540744819711, −3.39209034924814448431236710173, −1.65460087447422009133068963874, −0.66411484865261720987435053928, 1.85537870783315910909613390404, 2.63051155307174455705926928247, 4.24014062089085409062533311428, 5.19101849291672779618847231035, 6.25238791389427180963692138518, 7.10264854400192369681046855088, 8.493389194201133996954941135750, 9.408286567295937696282307019183, 10.83069785838860819413599602183, 11.77578118933270374031583537979

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