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Properties

Label 2-180-180.119-c1-0-18
Degree $2$
Conductor $180$
Sign $0.535 + 0.844i$
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 + 0.288i)2-s + (0.112 − 1.72i)3-s + (1.83 − 0.799i)4-s + (2.23 + 0.145i)5-s + (0.343 + 2.42i)6-s + (0.573 − 0.993i)7-s + (−2.30 + 1.63i)8-s + (−2.97 − 0.388i)9-s + (−3.13 + 0.442i)10-s + (−0.629 + 1.09i)11-s + (−1.17 − 3.25i)12-s + (4.39 − 2.53i)13-s + (−0.507 + 1.54i)14-s + (0.502 − 3.84i)15-s + (2.72 − 2.93i)16-s − 3.29·17-s + ⋯
L(s)  = 1  + (−0.978 + 0.204i)2-s + (0.0649 − 0.997i)3-s + (0.916 − 0.399i)4-s + (0.997 + 0.0650i)5-s + (0.140 + 0.990i)6-s + (0.216 − 0.375i)7-s + (−0.815 + 0.578i)8-s + (−0.991 − 0.129i)9-s + (−0.990 + 0.140i)10-s + (−0.189 + 0.328i)11-s + (−0.339 − 0.940i)12-s + (1.21 − 0.704i)13-s + (−0.135 + 0.411i)14-s + (0.129 − 0.991i)15-s + (0.680 − 0.732i)16-s − 0.799·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.535 + 0.844i$
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: \((2,\ 180,\ (\ :1/2),\ 0.535 + 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.796906 - 0.438272i\)
\(L(\frac12)\) \(\approx\) \(0.796906 - 0.438272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.288i)T \)
3 \( 1 + (-0.112 + 1.72i)T \)
5 \( 1 + (-2.23 - 0.145i)T \)
good7 \( 1 + (-0.573 + 0.993i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.629 - 1.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.39 + 2.53i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.29T + 17T^{2} \)
19 \( 1 + 3.62iT - 19T^{2} \)
23 \( 1 + (-3.09 + 1.78i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.184 + 0.106i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (9.12 - 5.26i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.72iT - 37T^{2} \)
41 \( 1 + (-5.81 + 3.35i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.41 - 2.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.925 - 0.534i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.12T + 53T^{2} \)
59 \( 1 + (-4.87 - 8.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.24 - 9.08i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.42 - 12.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.68T + 71T^{2} \)
73 \( 1 + 9.08iT - 73T^{2} \)
79 \( 1 + (-5.78 - 3.34i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.93 + 2.84i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.26iT - 89T^{2} \)
97 \( 1 + (6.69 + 3.86i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.64115329690196143934342072175, −11.13020181706825472593789710583, −10.65679825842572817047313279275, −9.222611363589394391771124273627, −8.515083538088475015030255897628, −7.29558863072038473192591486622, −6.49590780450420466030783857338, −5.45671192865565409570186183055, −2.71339816879458631965280979967, −1.30353190044559788074834591038, 2.06371143888258996022321316357, 3.66810492556379984272972916317, 5.49415871629392225202809924825, 6.45258222887065806381319872141, 8.178285644978688711666469813843, 9.111916693749015779679292101725, 9.579163592299640778581659868291, 10.93736601646707087727399791251, 11.15117034371199337707618348430, 12.69946328624380798967187714782

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