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Properties

Label 2-180-12.11-c1-0-4
Degree $2$
Conductor $180$
Sign $0.948 - 0.316i$
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  + (1.39 + 0.207i)2-s + (1.91 + 0.579i)4-s + i·5-s − 1.63i·7-s + (2.55 + 1.20i)8-s + (−0.207 + 1.39i)10-s − 3.95·11-s + 0.585·13-s + (0.339 − 2.29i)14-s + (3.32 + 2.21i)16-s + 4i·17-s − 7.91i·19-s + (−0.579 + 1.91i)20-s + (−5.53 − 0.819i)22-s − 5.59·23-s + ⋯
L(s)  = 1  + (0.989 + 0.146i)2-s + (0.957 + 0.289i)4-s + 0.447i·5-s − 0.619i·7-s + (0.904 + 0.426i)8-s + (−0.0654 + 0.442i)10-s − 1.19·11-s + 0.162·13-s + (0.0907 − 0.612i)14-s + (0.832 + 0.554i)16-s + 0.970i·17-s − 1.81i·19-s + (−0.129 + 0.428i)20-s + (−1.18 − 0.174i)22-s − 1.16·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.948 - 0.316i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.948 - 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93292 + 0.313450i\)
\(L(\frac12)\) \(\approx\) \(1.93292 + 0.313450i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.207i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 + 1.63iT - 7T^{2} \)
11 \( 1 + 3.95T + 11T^{2} \)
13 \( 1 - 0.585T + 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 + 7.91iT - 19T^{2} \)
23 \( 1 + 5.59T + 23T^{2} \)
29 \( 1 + 7.65iT - 29T^{2} \)
31 \( 1 - 5.59iT - 31T^{2} \)
37 \( 1 + 9.07T + 37T^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 - 7.91iT - 43T^{2} \)
47 \( 1 - 3.27T + 47T^{2} \)
53 \( 1 + 5.17iT - 53T^{2} \)
59 \( 1 - 11.8T + 59T^{2} \)
61 \( 1 + 0.828T + 61T^{2} \)
67 \( 1 - 11.1iT - 67T^{2} \)
71 \( 1 - 4.63T + 71T^{2} \)
73 \( 1 - 8.82T + 73T^{2} \)
79 \( 1 + 10.2iT - 79T^{2} \)
83 \( 1 + 3.27T + 83T^{2} \)
89 \( 1 - 5.41iT - 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−13.01905880159703169637533254224, −11.80363489807609788414993252424, −10.82507902261869814958244209425, −10.16716374651235624086980893871, −8.316152661136245216062297616748, −7.32680206645652711533342789541, −6.33925612888562548924033464434, −5.09487052258034031005307469927, −3.85890993164528106041357574452, −2.46688580335935611625897960926, 2.16131970122140972135150112451, 3.68035551267735984724410978728, 5.16573032951059718213932810305, 5.82609118789874247891310929020, 7.33847167614032176260250186326, 8.404075718623126243612023228183, 9.880678679792927513911870890119, 10.77105114847513772154306569578, 12.07321306876568230508602276479, 12.43784262015834180964243973029

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