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Properties

Label 2-180-180.43-c1-0-2
Degree $2$
Conductor $180$
Sign $-0.969 - 0.246i$
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  + (0.557 + 1.29i)2-s + (−1.15 − 1.29i)3-s + (−1.37 + 1.44i)4-s + (−1.50 + 1.65i)5-s + (1.04 − 2.21i)6-s + (−3.69 + 0.989i)7-s + (−2.65 − 0.985i)8-s + (−0.348 + 2.97i)9-s + (−2.98 − 1.03i)10-s + (3.31 + 1.91i)11-s + (3.46 + 0.116i)12-s + (−0.496 + 1.85i)13-s + (−3.34 − 4.25i)14-s + (3.87 + 0.0432i)15-s + (−0.197 − 3.99i)16-s + (1.83 − 1.83i)17-s + ⋯
L(s)  = 1  + (0.394 + 0.919i)2-s + (−0.664 − 0.747i)3-s + (−0.689 + 0.724i)4-s + (−0.673 + 0.739i)5-s + (0.424 − 0.905i)6-s + (−1.39 + 0.374i)7-s + (−0.937 − 0.348i)8-s + (−0.116 + 0.993i)9-s + (−0.944 − 0.327i)10-s + (0.998 + 0.576i)11-s + (0.999 + 0.0335i)12-s + (−0.137 + 0.514i)13-s + (−0.894 − 1.13i)14-s + (0.999 + 0.0111i)15-s + (−0.0492 − 0.998i)16-s + (0.444 − 0.444i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.969 - 0.246i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ -0.969 - 0.246i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0712889 + 0.569067i\)
\(L(\frac12)\) \(\approx\) \(0.0712889 + 0.569067i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.557 - 1.29i)T \)
3 \( 1 + (1.15 + 1.29i)T \)
5 \( 1 + (1.50 - 1.65i)T \)
good7 \( 1 + (3.69 - 0.989i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3.31 - 1.91i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.496 - 1.85i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-1.83 + 1.83i)T - 17iT^{2} \)
19 \( 1 + 3.60T + 19T^{2} \)
23 \( 1 + (-0.0567 - 0.0151i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-5.91 - 3.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.34 - 1.92i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.46 - 3.46i)T - 37iT^{2} \)
41 \( 1 + (-1.56 - 2.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.17 - 11.8i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (5.46 - 1.46i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.82 + 2.82i)T + 53iT^{2} \)
59 \( 1 + (-4.94 - 8.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.42 - 4.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.406 + 1.51i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.70iT - 71T^{2} \)
73 \( 1 + (6.08 + 6.08i)T + 73iT^{2} \)
79 \( 1 + (-6.64 + 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.845 + 3.15i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 9.47iT - 89T^{2} \)
97 \( 1 + (2.18 + 8.14i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.98187246496367155462013147548, −12.29435132793346475228889901901, −11.59440762365558525053008502799, −10.07363016248562533521938000766, −8.868592871916460372660210026962, −7.51758170887556927649113736033, −6.65666633044957267004798633090, −6.24188818400748245332138220838, −4.56310323004131019379922700527, −3.09131467836750967038559843186, 0.49944491864644134535346796788, 3.49074046443068062478957705996, 4.10492467581139314564320920119, 5.50948510350931466045567322828, 6.54262429583646764286742415021, 8.610872907778098281329124883695, 9.491274366899960995600031662065, 10.34511807544853083871222085032, 11.26394053270286795492985100500, 12.31225041594026616809610322378

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