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Properties

Label 2-180-12.11-c1-0-5
Degree $2$
Conductor $180$
Sign $0.886 + 0.462i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.736 + 1.20i)2-s + (−0.914 − 1.77i)4-s − i·5-s − 5.03i·7-s + (2.82 + 0.207i)8-s + (1.20 + 0.736i)10-s − 2.08·11-s + 3.41·13-s + (6.07 + 3.70i)14-s + (−2.32 + 3.25i)16-s − 4i·17-s + 4.16i·19-s + (−1.77 + 0.914i)20-s + (1.53 − 2.51i)22-s + 2.94·23-s + ⋯
L(s)  = 1  + (−0.521 + 0.853i)2-s + (−0.457 − 0.889i)4-s − 0.447i·5-s − 1.90i·7-s + (0.997 + 0.0732i)8-s + (0.381 + 0.233i)10-s − 0.628·11-s + 0.946·13-s + (1.62 + 0.990i)14-s + (−0.582 + 0.813i)16-s − 0.970i·17-s + 0.956i·19-s + (−0.397 + 0.204i)20-s + (0.327 − 0.536i)22-s + 0.614·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.886 + 0.462i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.886 + 0.462i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.803218 - 0.196806i\)
\(L(\frac12)\) \(\approx\) \(0.803218 - 0.196806i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.736 - 1.20i)T \)
3 \( 1 \)
5 \( 1 + iT \)
good7 \( 1 + 5.03iT - 7T^{2} \)
11 \( 1 + 2.08T + 11T^{2} \)
13 \( 1 - 3.41T + 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - 4.16iT - 19T^{2} \)
23 \( 1 - 2.94T + 23T^{2} \)
29 \( 1 + 3.65iT - 29T^{2} \)
31 \( 1 - 2.94iT - 31T^{2} \)
37 \( 1 - 5.07T + 37T^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + 4.16iT - 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 - 10.8iT - 53T^{2} \)
59 \( 1 - 6.25T + 59T^{2} \)
61 \( 1 - 4.82T + 61T^{2} \)
67 \( 1 - 5.89iT - 67T^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 - 3.17T + 73T^{2} \)
79 \( 1 - 11.2iT - 79T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 + 2.58iT - 89T^{2} \)
97 \( 1 + 2.48T + 97T^{2} \)
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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

−13.02713038982029294241379892905, −11.24586314285710942637314540763, −10.40440501860480037119311179926, −9.586517994894482867076593071795, −8.263976265525217769286572009531, −7.51665825848995621809422346512, −6.50932043899934503554129328850, −5.10996689154766058152743004267, −3.94811917144873871088255721478, −0.977227666693878819353306141720, 2.17022206156719609799753510098, 3.29336171064823203356279132584, 5.11049119843668401458977973289, 6.43088035438489143208292240944, 8.079919681238110733364216659705, 8.785322922763669473499195183002, 9.713577976633292254331708407949, 10.97520497812079683340827138642, 11.51419099881000100870290991762, 12.68230458553712474932779714068

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