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Properties

Label 2-180-4.3-c2-0-18
Degree $2$
Conductor $180$
Sign $-0.309 + 0.951i$
Analytic cond. $4.90464$
Root an. cond. $2.21464$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.61 − 1.17i)2-s + (1.23 − 3.80i)4-s − 2.23·5-s − 8.50i·7-s + (−2.47 − 7.60i)8-s + (−3.61 + 2.62i)10-s − 1.79i·11-s + 0.472·13-s + (−10 − 13.7i)14-s + (−12.9 − 9.40i)16-s + 23.8·17-s − 9.40i·19-s + (−2.76 + 8.50i)20-s + (−2.11 − 2.90i)22-s + 16.1i·23-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s − 0.447·5-s − 1.21i·7-s + (−0.309 − 0.951i)8-s + (−0.361 + 0.262i)10-s − 0.163i·11-s + 0.0363·13-s + (−0.714 − 0.983i)14-s + (−0.809 − 0.587i)16-s + 1.40·17-s − 0.494i·19-s + (−0.138 + 0.425i)20-s + (−0.0959 − 0.132i)22-s + 0.700i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 + 0.951i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.309 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.309 + 0.951i$
Analytic conductor: \(4.90464\)
Root analytic conductor: \(2.21464\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1),\ -0.309 + 0.951i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.22293 - 1.68321i\)
\(L(\frac12)\) \(\approx\) \(1.22293 - 1.68321i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.61 + 1.17i)T \)
3 \( 1 \)
5 \( 1 + 2.23T \)
good7 \( 1 + 8.50iT - 49T^{2} \)
11 \( 1 + 1.79iT - 121T^{2} \)
13 \( 1 - 0.472T + 169T^{2} \)
17 \( 1 - 23.8T + 289T^{2} \)
19 \( 1 + 9.40iT - 361T^{2} \)
23 \( 1 - 16.1iT - 529T^{2} \)
29 \( 1 + 6.94T + 841T^{2} \)
31 \( 1 - 47.4iT - 961T^{2} \)
37 \( 1 - 26.3T + 1.36e3T^{2} \)
41 \( 1 - 41.4T + 1.68e3T^{2} \)
43 \( 1 + 2.00iT - 1.84e3T^{2} \)
47 \( 1 - 35.3iT - 2.20e3T^{2} \)
53 \( 1 - 21.6T + 2.80e3T^{2} \)
59 \( 1 + 73.8iT - 3.48e3T^{2} \)
61 \( 1 + 26.1T + 3.72e3T^{2} \)
67 \( 1 - 88.8iT - 4.48e3T^{2} \)
71 \( 1 - 39.4iT - 5.04e3T^{2} \)
73 \( 1 - 137.T + 5.32e3T^{2} \)
79 \( 1 + 113. iT - 6.24e3T^{2} \)
83 \( 1 + 21.2iT - 6.88e3T^{2} \)
89 \( 1 + 67.4T + 7.92e3T^{2} \)
97 \( 1 + 39.1T + 9.40e3T^{2} \)
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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

−12.17935458242749947502053338363, −11.18489475553934111227339603006, −10.44816809918389292977110307322, −9.451583086160774726249118546747, −7.79653319208947233388284032214, −6.82252430635011132095208689330, −5.41885683525770648456837903213, −4.18585968978765290348821271261, −3.18716029263427950265925992322, −1.06184625448812135752651233757, 2.57717710395715256225166291396, 3.96119809043752482827936122196, 5.34113970026599315939787309665, 6.17013446208395822952469741981, 7.57315541498131009025042517263, 8.357881654070276076538388048451, 9.552779596615961171881707755653, 11.11600710502212103317796420764, 12.14531160836601808451197827819, 12.51666769876979165498290159691

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