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Properties

Label 2-180-20.19-c2-0-24
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

\(L(\frac{3}{2})\) \(\approx\) \(2.27889 - 1.34357i\)
\(L(\frac12)\) \(\approx\) \(2.27889 - 1.34357i\)
\(L(2)\) 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

−12.39036038057723585907301284774, −11.59917555545583085509113575821, −10.44126539596657512392197842312, −9.375355082067287585024404553330, −8.099139767922506522851436855834, −6.81560877430000993371432261209, −5.50147810588176706719404931623, −4.67387305211446898093836168124, −3.28431358399553533355316164406, −1.43094506493294664169337786972, 2.42700851535336932045025801689, 3.63575559246780927556807195483, 5.01489267642542554817514436976, 6.34282455026921302690943712313, 7.07039562035601777884115027117, 8.219653089388856729165542036522, 9.780238479851717673392350052165, 11.11063590027680867369194943735, 11.41005890468472452599889020734, 12.93305225283570481195562584663

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