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Properties

Label 2-180-180.103-c1-0-19
Degree $2$
Conductor $180$
Sign $0.876 - 0.481i$
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  + (1.41 + 0.0886i)2-s + (0.119 + 1.72i)3-s + (1.98 + 0.250i)4-s + (0.395 − 2.20i)5-s + (0.0147 + 2.44i)6-s + (0.331 − 1.23i)7-s + (2.77 + 0.529i)8-s + (−2.97 + 0.411i)9-s + (0.753 − 3.07i)10-s + (−4.09 + 2.36i)11-s + (−0.196 + 3.45i)12-s + (−1.67 + 0.448i)13-s + (0.576 − 1.71i)14-s + (3.84 + 0.421i)15-s + (3.87 + 0.993i)16-s + (−2.37 + 2.37i)17-s + ⋯
L(s)  = 1  + (0.998 + 0.0627i)2-s + (0.0687 + 0.997i)3-s + (0.992 + 0.125i)4-s + (0.176 − 0.984i)5-s + (0.00602 + 0.999i)6-s + (0.125 − 0.466i)7-s + (0.982 + 0.187i)8-s + (−0.990 + 0.137i)9-s + (0.238 − 0.971i)10-s + (−1.23 + 0.712i)11-s + (−0.0566 + 0.998i)12-s + (−0.464 + 0.124i)13-s + (0.154 − 0.458i)14-s + (0.994 + 0.108i)15-s + (0.968 + 0.248i)16-s + (−0.575 + 0.575i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.876 - 0.481i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.876 - 0.481i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94651 + 0.499464i\)
\(L(\frac12)\) \(\approx\) \(1.94651 + 0.499464i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0886i)T \)
3 \( 1 + (-0.119 - 1.72i)T \)
5 \( 1 + (-0.395 + 2.20i)T \)
good7 \( 1 + (-0.331 + 1.23i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (4.09 - 2.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.67 - 0.448i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (2.37 - 2.37i)T - 17iT^{2} \)
19 \( 1 - 2.26T + 19T^{2} \)
23 \( 1 + (2.26 + 8.43i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.111 + 0.0641i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.75 - 3.32i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.433 - 0.433i)T - 37iT^{2} \)
41 \( 1 + (-3.06 + 5.30i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.10 - 1.09i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.24 - 4.63i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.98 + 3.98i)T + 53iT^{2} \)
59 \( 1 + (7.36 - 12.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.25 - 9.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.219 + 0.0587i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 3.57iT - 71T^{2} \)
73 \( 1 + (4.90 + 4.90i)T + 73iT^{2} \)
79 \( 1 + (0.916 + 1.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.78 + 1.55i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 11.4iT - 89T^{2} \)
97 \( 1 + (3.00 + 0.803i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.76727244653171548122042881401, −11.99558520175231917788616629683, −10.65529575467694970307211925929, −10.09806242778534905841855865242, −8.663078346789631625915283309067, −7.58655227478441211989420626704, −5.97454499602810683255147598445, −4.80627262237121831503839080719, −4.32669142827459589626478780217, −2.53240529514521570954993350917, 2.29719661482523669203508897534, 3.18300295556892615361656163630, 5.31128602865918897915612458804, 6.11949976269476116851616941653, 7.27334058744972589849015041061, 7.993847969258435415786137272432, 9.813923541716619754725221893891, 11.14555326226613215293822782062, 11.60746960217565030123359079063, 12.75446571705732086724055248662

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