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Properties

Label 2-180-4.3-c6-0-44
Degree $2$
Conductor $180$
Sign $-0.789 + 0.613i$
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  + (−2.59 − 7.56i)2-s + (−50.5 + 39.2i)4-s + 55.9·5-s − 671. i·7-s + (428. + 280. i)8-s + (−144. − 423. i)10-s − 199. i·11-s + 2.04e3·13-s + (−5.07e3 + 1.74e3i)14-s + (1.01e3 − 3.96e3i)16-s + 8.21e3·17-s + 5.14e3i·19-s + (−2.82e3 + 2.19e3i)20-s + (−1.51e3 + 517. i)22-s + 1.46e4i·23-s + ⋯
L(s)  = 1  + (−0.324 − 0.945i)2-s + (−0.789 + 0.613i)4-s + 0.447·5-s − 1.95i·7-s + (0.836 + 0.548i)8-s + (−0.144 − 0.423i)10-s − 0.149i·11-s + 0.929·13-s + (−1.85 + 0.634i)14-s + (0.247 − 0.968i)16-s + 1.67·17-s + 0.750i·19-s + (−0.353 + 0.274i)20-s + (−0.141 + 0.0486i)22-s + 1.20i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 + 0.613i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.789 + 0.613i$
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: \((2,\ 180,\ (\ :3),\ -0.789 + 0.613i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.794517446\)
\(L(\frac12)\) \(\approx\) \(1.794517446\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.59 + 7.56i)T \)
3 \( 1 \)
5 \( 1 - 55.9T \)
good7 \( 1 + 671. iT - 1.17e5T^{2} \)
11 \( 1 + 199. iT - 1.77e6T^{2} \)
13 \( 1 - 2.04e3T + 4.82e6T^{2} \)
17 \( 1 - 8.21e3T + 2.41e7T^{2} \)
19 \( 1 - 5.14e3iT - 4.70e7T^{2} \)
23 \( 1 - 1.46e4iT - 1.48e8T^{2} \)
29 \( 1 - 1.25e4T + 5.94e8T^{2} \)
31 \( 1 + 5.44e4iT - 8.87e8T^{2} \)
37 \( 1 + 11.6T + 2.56e9T^{2} \)
41 \( 1 - 6.78e4T + 4.75e9T^{2} \)
43 \( 1 + 1.05e5iT - 6.32e9T^{2} \)
47 \( 1 + 8.96e4iT - 1.07e10T^{2} \)
53 \( 1 - 1.36e3T + 2.21e10T^{2} \)
59 \( 1 - 5.60e3iT - 4.21e10T^{2} \)
61 \( 1 + 1.32e5T + 5.15e10T^{2} \)
67 \( 1 - 3.25e5iT - 9.04e10T^{2} \)
71 \( 1 + 3.03e5iT - 1.28e11T^{2} \)
73 \( 1 + 2.47e5T + 1.51e11T^{2} \)
79 \( 1 + 3.89e5iT - 2.43e11T^{2} \)
83 \( 1 - 2.15e5iT - 3.26e11T^{2} \)
89 \( 1 + 4.62e4T + 4.96e11T^{2} \)
97 \( 1 - 8.00e5T + 8.32e11T^{2} \)
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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

−10.95867979191140148868212123019, −10.23582980213827612946782550627, −9.572082098408170192686812148302, −8.094689622399135967185897109993, −7.35336115049333899548452391277, −5.70358997227247419857011881444, −4.11195428627647898774800217748, −3.39461802270166051610650172988, −1.50727167032350587622972103998, −0.67564104885978514090330819436, 1.27487402037333568296620119214, 2.89028208785602722518223247169, 4.88608314037966913027386790584, 5.77862595221615245742105348300, 6.52194739026243239390627361819, 8.070310172351104994005698423889, 8.832358328100122666961955960023, 9.578604312609149457998034815299, 10.75092556054698093469660338175, 12.16660938925237434062469496980

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