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Properties

Label 2-180-20.19-c2-0-4
Degree $2$
Conductor $180$
Sign $-0.484 - 0.875i$
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  + (−1.93 + 0.5i)2-s + (3.50 − 1.93i)4-s + 5i·5-s + (−5.80 + 5.50i)8-s + (−2.5 − 9.68i)10-s + (8.50 − 13.5i)16-s + 14i·17-s + 30.9i·19-s + (9.68 + 17.5i)20-s − 30.9·23-s − 25·25-s + 61.9i·31-s + (−9.68 + 30.5i)32-s + (−7 − 27.1i)34-s + (−15.4 − 60.0i)38-s + ⋯
L(s)  = 1  + (−0.968 + 0.250i)2-s + (0.875 − 0.484i)4-s + i·5-s + (−0.726 + 0.687i)8-s + (−0.250 − 0.968i)10-s + (0.531 − 0.847i)16-s + 0.823i·17-s + 1.63i·19-s + (0.484 + 0.875i)20-s − 1.34·23-s − 25-s + 1.99i·31-s + (−0.302 + 0.953i)32-s + (−0.205 − 0.797i)34-s + (−0.407 − 1.57i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.875i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.484 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.484 - 0.875i$
Analytic conductor: \(4.90464\)
Root analytic conductor: \(2.21464\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1),\ -0.484 - 0.875i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.380434 + 0.645269i\)
\(L(\frac12)\) \(\approx\) \(0.380434 + 0.645269i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.93 - 0.5i)T \)
3 \( 1 \)
5 \( 1 - 5iT \)
good7 \( 1 + 49T^{2} \)
11 \( 1 - 121T^{2} \)
13 \( 1 - 169T^{2} \)
17 \( 1 - 14iT - 289T^{2} \)
19 \( 1 - 30.9iT - 361T^{2} \)
23 \( 1 + 30.9T + 529T^{2} \)
29 \( 1 + 841T^{2} \)
31 \( 1 - 61.9iT - 961T^{2} \)
37 \( 1 - 1.36e3T^{2} \)
41 \( 1 + 1.68e3T^{2} \)
43 \( 1 + 1.84e3T^{2} \)
47 \( 1 - 92.9T + 2.20e3T^{2} \)
53 \( 1 + 86iT - 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 - 118T + 3.72e3T^{2} \)
67 \( 1 + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 5.32e3T^{2} \)
79 \( 1 + 123. iT - 6.24e3T^{2} \)
83 \( 1 + 61.9T + 6.88e3T^{2} \)
89 \( 1 + 7.92e3T^{2} \)
97 \( 1 - 9.40e3T^{2} \)
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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

−12.46254535910019317292632806460, −11.54034235928798549631021202882, −10.40192219566774542786132363234, −10.04689154720564641590874989529, −8.577556139829743073259236462164, −7.71511614337214403637906438029, −6.62451444504420816018397345711, −5.73481952696108318749166615691, −3.55505473700686498126572765941, −1.91892529519781189932154374780, 0.59616570877129900726359977756, 2.38899370000382688648787484083, 4.24590222009882136597292985208, 5.78794047065834910712782675606, 7.19612007305331832428950264862, 8.191772253089824194086351723127, 9.170748296611558363720919005926, 9.823283701104276100675020113378, 11.17017430144331246600640766473, 11.89341782707607827539098145182

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