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Properties

Label 2-180-12.11-c5-0-31
Degree $2$
Conductor $180$
Sign $-0.887 + 0.461i$
Analytic cond. $28.8690$
Root an. cond. $5.37299$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.384 − 5.64i)2-s + (−31.7 − 4.34i)4-s + 25i·5-s − 166. i·7-s + (−36.6 + 177. i)8-s + (141. + 9.61i)10-s + 775.·11-s + 300.·13-s + (−937. − 63.8i)14-s + (986. + 275. i)16-s − 1.15e3i·17-s + 1.79e3i·19-s + (108. − 792. i)20-s + (298. − 4.37e3i)22-s − 3.27e3·23-s + ⋯
L(s)  = 1  + (0.0679 − 0.997i)2-s + (−0.990 − 0.135i)4-s + 0.447i·5-s − 1.28i·7-s + (−0.202 + 0.979i)8-s + (0.446 + 0.0303i)10-s + 1.93·11-s + 0.493·13-s + (−1.27 − 0.0870i)14-s + (0.963 + 0.268i)16-s − 0.969i·17-s + 1.14i·19-s + (0.0606 − 0.443i)20-s + (0.131 − 1.92i)22-s − 1.28·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.887 + 0.461i$
Analytic conductor: \(28.8690\)
Root analytic conductor: \(5.37299\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :5/2),\ -0.887 + 0.461i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.654173063\)
\(L(\frac12)\) \(\approx\) \(1.654173063\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.384 + 5.64i)T \)
3 \( 1 \)
5 \( 1 - 25iT \)
good7 \( 1 + 166. iT - 1.68e4T^{2} \)
11 \( 1 - 775.T + 1.61e5T^{2} \)
13 \( 1 - 300.T + 3.71e5T^{2} \)
17 \( 1 + 1.15e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.79e3iT - 2.47e6T^{2} \)
23 \( 1 + 3.27e3T + 6.43e6T^{2} \)
29 \( 1 + 7.78e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.57e3iT - 2.86e7T^{2} \)
37 \( 1 + 4.97e3T + 6.93e7T^{2} \)
41 \( 1 + 1.59e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.35e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.55e4T + 2.29e8T^{2} \)
53 \( 1 - 3.08e3iT - 4.18e8T^{2} \)
59 \( 1 - 4.97e4T + 7.14e8T^{2} \)
61 \( 1 - 2.22e4T + 8.44e8T^{2} \)
67 \( 1 - 4.79e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.74e4T + 1.80e9T^{2} \)
73 \( 1 + 5.31e4T + 2.07e9T^{2} \)
79 \( 1 + 4.78e4iT - 3.07e9T^{2} \)
83 \( 1 - 1.73e4T + 3.93e9T^{2} \)
89 \( 1 + 4.99e4iT - 5.58e9T^{2} \)
97 \( 1 - 5.79e4T + 8.58e9T^{2} \)
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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

−11.54137564370475126025750055197, −10.30221299799005978493834591615, −9.734974909416201847909609713389, −8.492875090227618397352124579141, −7.19739178199761916965502638136, −5.97036575818015750843339705061, −4.12821741260666920272588362337, −3.68516835985561937358861604139, −1.81944880520628795347579983977, −0.56887861044757300774636382169, 1.41615678135976620585885475429, 3.58875535369400770690117962877, 4.82366689144806517064351881165, 6.02938544696040856128947488676, 6.71440458533338500343022637384, 8.365513325623096523486250948085, 8.866437328798361348221591892005, 9.715845570564104908728986976813, 11.46437365538394091291986726400, 12.33647916905433186340687595503

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