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Properties

Label 2-180-60.23-c2-0-19
Degree $2$
Conductor $180$
Sign $-0.179 + 0.983i$
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  + (0.537 + 1.92i)2-s + (−3.42 + 2.06i)4-s + (−2.15 − 4.51i)5-s + (−4.16 − 4.16i)7-s + (−5.82 − 5.48i)8-s + (7.54 − 6.56i)10-s − 21.7·11-s + (3.43 + 3.43i)13-s + (5.78 − 10.2i)14-s + (7.43 − 14.1i)16-s + (−13.0 − 13.0i)17-s + 19.7·19-s + (16.7 + 11.0i)20-s + (−11.6 − 41.9i)22-s + (17.1 + 17.1i)23-s + ⋯
L(s)  = 1  + (0.268 + 0.963i)2-s + (−0.855 + 0.517i)4-s + (−0.430 − 0.902i)5-s + (−0.594 − 0.594i)7-s + (−0.728 − 0.685i)8-s + (0.754 − 0.656i)10-s − 1.97·11-s + (0.264 + 0.264i)13-s + (0.413 − 0.732i)14-s + (0.464 − 0.885i)16-s + (−0.767 − 0.767i)17-s + 1.03·19-s + (0.835 + 0.550i)20-s + (−0.531 − 1.90i)22-s + (0.743 + 0.743i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.179 + 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.179 + 0.983i$
Analytic conductor: \(4.90464\)
Root analytic conductor: \(2.21464\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1),\ -0.179 + 0.983i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.241700 - 0.289886i\)
\(L(\frac12)\) \(\approx\) \(0.241700 - 0.289886i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.537 - 1.92i)T \)
3 \( 1 \)
5 \( 1 + (2.15 + 4.51i)T \)
good7 \( 1 + (4.16 + 4.16i)T + 49iT^{2} \)
11 \( 1 + 21.7T + 121T^{2} \)
13 \( 1 + (-3.43 - 3.43i)T + 169iT^{2} \)
17 \( 1 + (13.0 + 13.0i)T + 289iT^{2} \)
19 \( 1 - 19.7T + 361T^{2} \)
23 \( 1 + (-17.1 - 17.1i)T + 529iT^{2} \)
29 \( 1 + 20.8T + 841T^{2} \)
31 \( 1 + 12.9iT - 961T^{2} \)
37 \( 1 + (21.0 - 21.0i)T - 1.36e3iT^{2} \)
41 \( 1 + 58.0iT - 1.68e3T^{2} \)
43 \( 1 + (11.2 - 11.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (4.90 - 4.90i)T - 2.20e3iT^{2} \)
53 \( 1 + (52.7 - 52.7i)T - 2.80e3iT^{2} \)
59 \( 1 + 72.9iT - 3.48e3T^{2} \)
61 \( 1 - 2.48T + 3.72e3T^{2} \)
67 \( 1 + (0.141 + 0.141i)T + 4.48e3iT^{2} \)
71 \( 1 + 16.7T + 5.04e3T^{2} \)
73 \( 1 + (28.6 + 28.6i)T + 5.32e3iT^{2} \)
79 \( 1 + 111.T + 6.24e3T^{2} \)
83 \( 1 + (-99.0 - 99.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 47.0T + 7.92e3T^{2} \)
97 \( 1 + (-49.4 + 49.4i)T - 9.40e3iT^{2} \)
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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

−12.57087556024491556034162685994, −11.28069342435312403711526673005, −9.867495804550926127182484447306, −8.936996629804878776884124520745, −7.80074920730372764254656495777, −7.13975761949688160584443210812, −5.54396726525615958697525819951, −4.74913188798492732260555287755, −3.33174738619211621137901158304, −0.19690213006866818465244202301, 2.50913372303506341699906715921, 3.35808986507978360796935762522, 5.00794389846791223912161927269, 6.16326408063781655359540922555, 7.69469736108558190458548125754, 8.829602242815632390982021205982, 10.13820937634531936201628681143, 10.71946275133329164601257329613, 11.61392111070846241019114411938, 12.81993271685713434011326837465

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