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Properties

Label 2-180-20.19-c2-0-18
Degree $2$
Conductor $180$
Sign $0.866 + 0.5i$
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  + (1.73 − i)2-s + (1.99 − 3.46i)4-s + 5i·5-s + 10.3·7-s − 7.99i·8-s + (5 + 8.66i)10-s + 10.3i·11-s − 18i·13-s + (18 − 10.3i)14-s + (−8 − 13.8i)16-s − 10i·17-s + 13.8i·19-s + (17.3 + 9.99i)20-s + (10.3 + 18i)22-s + 6.92·23-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s + 1.48·7-s − 0.999i·8-s + (0.5 + 0.866i)10-s + 0.944i·11-s − 1.38i·13-s + (1.28 − 0.742i)14-s + (−0.5 − 0.866i)16-s − 0.588i·17-s + 0.729i·19-s + (0.866 + 0.499i)20-s + (0.472 + 0.818i)22-s + 0.301·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.866 + 0.5i$
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: \((2,\ 180,\ (\ :1),\ 0.866 + 0.5i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.57640 - 0.690345i\)
\(L(\frac12)\) \(\approx\) \(2.57640 - 0.690345i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 + i)T \)
3 \( 1 \)
5 \( 1 - 5iT \)
good7 \( 1 - 10.3T + 49T^{2} \)
11 \( 1 - 10.3iT - 121T^{2} \)
13 \( 1 + 18iT - 169T^{2} \)
17 \( 1 + 10iT - 289T^{2} \)
19 \( 1 - 13.8iT - 361T^{2} \)
23 \( 1 - 6.92T + 529T^{2} \)
29 \( 1 + 36T + 841T^{2} \)
31 \( 1 - 6.92iT - 961T^{2} \)
37 \( 1 - 54iT - 1.36e3T^{2} \)
41 \( 1 + 18T + 1.68e3T^{2} \)
43 \( 1 + 20.7T + 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 + 26iT - 2.80e3T^{2} \)
59 \( 1 - 31.1iT - 3.48e3T^{2} \)
61 \( 1 + 74T + 3.72e3T^{2} \)
67 \( 1 - 41.5T + 4.48e3T^{2} \)
71 \( 1 + 103. iT - 5.04e3T^{2} \)
73 \( 1 + 36iT - 5.32e3T^{2} \)
79 \( 1 - 90.0iT - 6.24e3T^{2} \)
83 \( 1 + 90.0T + 6.88e3T^{2} \)
89 \( 1 + 18T + 7.92e3T^{2} \)
97 \( 1 + 72iT - 9.40e3T^{2} \)
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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

−12.23271500728991841655220310917, −11.39456525310650814281932209332, −10.62304976643558234992994736044, −9.817975002556394799668516878159, −7.980647666964529344070320259314, −7.08013393255983853849587827735, −5.65597396592379580872745693064, −4.67170285236554176737861702631, −3.20545555520036530562574252121, −1.80905440824910575948112813922, 1.84468990945888461962682378871, 3.99394460265033085792277533667, 4.89679929304748720896652796879, 5.86733273706036309341186507352, 7.32608154982488439577829728401, 8.374771765451133634202781992725, 9.043730863199164944235358007324, 11.13503180218245867659654839371, 11.53145255317382563496270171015, 12.64369800940543272325967424995

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