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Properties

Label 2-180-180.119-c1-0-17
Degree $2$
Conductor $180$
Sign $0.653 - 0.756i$
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  + (0.937 + 1.05i)2-s + (0.959 − 1.44i)3-s + (−0.242 + 1.98i)4-s + (0.520 + 2.17i)5-s + (2.42 − 0.335i)6-s + (−0.550 + 0.953i)7-s + (−2.32 + 1.60i)8-s + (−1.15 − 2.76i)9-s + (−1.81 + 2.58i)10-s + (2.84 − 4.92i)11-s + (2.62 + 2.25i)12-s + (−2.07 + 1.19i)13-s + (−1.52 + 0.310i)14-s + (3.63 + 1.33i)15-s + (−3.88 − 0.963i)16-s + 1.29·17-s + ⋯
L(s)  = 1  + (0.662 + 0.748i)2-s + (0.554 − 0.832i)3-s + (−0.121 + 0.992i)4-s + (0.232 + 0.972i)5-s + (0.990 − 0.136i)6-s + (−0.208 + 0.360i)7-s + (−0.823 + 0.567i)8-s + (−0.385 − 0.922i)9-s + (−0.574 + 0.818i)10-s + (0.856 − 1.48i)11-s + (0.759 + 0.651i)12-s + (−0.575 + 0.332i)13-s + (−0.407 + 0.0830i)14-s + (0.938 + 0.345i)15-s + (−0.970 − 0.240i)16-s + 0.313·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.653 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.653 - 0.756i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.653 - 0.756i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65046 + 0.755226i\)
\(L(\frac12)\) \(\approx\) \(1.65046 + 0.755226i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.937 - 1.05i)T \)
3 \( 1 + (-0.959 + 1.44i)T \)
5 \( 1 + (-0.520 - 2.17i)T \)
good7 \( 1 + (0.550 - 0.953i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.84 + 4.92i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.07 - 1.19i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.29T + 17T^{2} \)
19 \( 1 + 2.78iT - 19T^{2} \)
23 \( 1 + (-1.82 + 1.05i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.51 + 3.18i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.78 - 1.60i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.51iT - 37T^{2} \)
41 \( 1 + (-6.35 + 3.66i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.25 - 3.90i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.09 + 4.67i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + (-3.66 - 6.34i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.48 + 4.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.85 + 3.20i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 - 9.86iT - 73T^{2} \)
79 \( 1 + (-0.975 - 0.563i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.98 - 5.76i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.16iT - 89T^{2} \)
97 \( 1 + (11.1 + 6.42i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.16990749663827694999738289092, −11.95221407839799587008484948235, −11.26343505475196010496615139266, −9.408633930531316616605062593355, −8.530497377032141745153586416290, −7.35240711311858648358814933871, −6.54880938517145048140562375757, −5.72676651114796417426308698177, −3.66163477018940472762135664882, −2.65253053749019205514568789221, 1.93132192699217220987332352541, 3.71698368905244619654690513542, 4.60450634806924547881130113177, 5.59175939729249934638970819774, 7.39403244705123332310193192665, 9.013340822350816462382028175224, 9.658183046416068967782356596646, 10.35774058005259350670111111102, 11.69140963184763047698237562531, 12.62957789172893591858432259617

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