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Properties

Label 2-180-36.23-c1-0-8
Degree $2$
Conductor $180$
Sign $0.224 + 0.974i$
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  + (−1.39 + 0.228i)2-s + (−1.60 + 0.654i)3-s + (1.89 − 0.638i)4-s + (−0.866 − 0.5i)5-s + (2.08 − 1.27i)6-s + (−1.04 + 0.604i)7-s + (−2.49 + 1.32i)8-s + (2.14 − 2.09i)9-s + (1.32 + 0.499i)10-s + (−1.52 − 2.63i)11-s + (−2.62 + 2.26i)12-s + (2.53 − 4.39i)13-s + (1.32 − 1.08i)14-s + (1.71 + 0.235i)15-s + (3.18 − 2.42i)16-s − 2.23i·17-s + ⋯
L(s)  = 1  + (−0.986 + 0.161i)2-s + (−0.925 + 0.377i)3-s + (0.947 − 0.319i)4-s + (−0.387 − 0.223i)5-s + (0.852 − 0.522i)6-s + (−0.395 + 0.228i)7-s + (−0.883 + 0.468i)8-s + (0.714 − 0.699i)9-s + (0.418 + 0.157i)10-s + (−0.458 − 0.794i)11-s + (−0.756 + 0.653i)12-s + (0.704 − 1.21i)13-s + (0.353 − 0.289i)14-s + (0.443 + 0.0608i)15-s + (0.795 − 0.605i)16-s − 0.543i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.224 + 0.974i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.224 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.301995 - 0.240210i\)
\(L(\frac12)\) \(\approx\) \(0.301995 - 0.240210i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.228i)T \)
3 \( 1 + (1.60 - 0.654i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
good7 \( 1 + (1.04 - 0.604i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.52 + 2.63i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.53 + 4.39i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.23iT - 17T^{2} \)
19 \( 1 + 2.53iT - 19T^{2} \)
23 \( 1 + (-3.48 + 6.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.84 - 2.79i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.83 + 1.63i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.3T + 37T^{2} \)
41 \( 1 + (-3.70 - 2.13i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.74 + 2.15i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.15 - 8.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.42iT - 53T^{2} \)
59 \( 1 + (-4.46 + 7.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.32 - 2.30i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.4 + 7.20i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.49T + 71T^{2} \)
73 \( 1 - 5.10T + 73T^{2} \)
79 \( 1 + (-3.26 + 1.88i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.286 - 0.495i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.40iT - 89T^{2} \)
97 \( 1 + (-5.82 - 10.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.23619311099343082907879183300, −10.95736423411843730372240756436, −10.76379229341776936453489688800, −9.418843570871049524177465600919, −8.530632240170701825879189233480, −7.30155886090526771118953639096, −6.13658774451167953309953221170, −5.20464624274609006513217368240, −3.20991824610853693482296168164, −0.54033191219750227748699871000, 1.74503619233606634857297064431, 3.87148168869610616094147303425, 5.74604030176034470355456075952, 6.94574757925027966391357398538, 7.50255672200843889076305292117, 8.919680677705595333888451773125, 10.07269147693200120013180069629, 10.87702099819686785886311634118, 11.70726285699456252218819067027, 12.50214303175188027828450481287

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