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Properties

Label 2-180-180.119-c1-0-4
Degree $2$
Conductor $180$
Sign $-0.546 - 0.837i$
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  + (−1.38 − 0.282i)2-s + (−0.959 + 1.44i)3-s + (1.84 + 0.782i)4-s + (0.520 + 2.17i)5-s + (1.73 − 1.72i)6-s + (0.550 − 0.953i)7-s + (−2.32 − 1.60i)8-s + (−1.15 − 2.76i)9-s + (−0.106 − 3.16i)10-s + (−2.84 + 4.92i)11-s + (−2.89 + 1.90i)12-s + (−2.07 + 1.19i)13-s + (−1.03 + 1.16i)14-s + (−3.63 − 1.33i)15-s + (2.77 + 2.88i)16-s + 1.29·17-s + ⋯
L(s)  = 1  + (−0.979 − 0.199i)2-s + (−0.554 + 0.832i)3-s + (0.920 + 0.391i)4-s + (0.232 + 0.972i)5-s + (0.709 − 0.705i)6-s + (0.208 − 0.360i)7-s + (−0.823 − 0.567i)8-s + (−0.385 − 0.922i)9-s + (−0.0337 − 0.999i)10-s + (−0.856 + 1.48i)11-s + (−0.835 + 0.549i)12-s + (−0.575 + 0.332i)13-s + (−0.275 + 0.311i)14-s + (−0.938 − 0.345i)15-s + (0.693 + 0.720i)16-s + 0.313·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.546 - 0.837i$
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: \((2,\ 180,\ (\ :1/2),\ -0.546 - 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.252364 + 0.465871i\)
\(L(\frac12)\) \(\approx\) \(0.252364 + 0.465871i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.282i)T \)
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

−12.56059769567533445653654260709, −11.67816158451400104400266745297, −10.70983823427483979165516748172, −10.04403354640595571234944706744, −9.502449071994059189351826847030, −7.79617059472070021999438112022, −6.99364829203458326229177594493, −5.69683594357798734048055684096, −4.04299415704868142659424287930, −2.39322825929143461365484692560, 0.68071631203025826059195104953, 2.43679723395988252471321289275, 5.34073333277018633375777426733, 5.88240363341890997328469390896, 7.38432126308978325177910212880, 8.255114997750242724608596161495, 9.019729909593221281253363167996, 10.40733439063953688599885074667, 11.29028476776705650008573437369, 12.20768557660144910601244314780

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