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Properties

Label 2-18e2-9.4-c3-0-8
Degree $2$
Conductor $324$
Sign $0.173 + 0.984i$
Analytic cond. $19.1166$
Root an. cond. $4.37225$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 − 2.59i)5-s + (2 + 3.46i)7-s + (−12 − 20.7i)11-s + (12.5 − 21.6i)13-s + 21·17-s − 52·19-s + (84 − 145. i)23-s + (58 + 100. i)25-s + (−88.5 − 153. i)29-s + (62 − 107. i)31-s + 12·35-s − 265·37-s + (213 − 368. i)41-s + (80 + 138. i)43-s + (−270 − 467. i)47-s + ⋯
L(s)  = 1  + (0.134 − 0.232i)5-s + (0.107 + 0.187i)7-s + (−0.328 − 0.569i)11-s + (0.266 − 0.461i)13-s + 0.299·17-s − 0.627·19-s + (0.761 − 1.31i)23-s + (0.464 + 0.803i)25-s + (−0.566 − 0.981i)29-s + (0.359 − 0.622i)31-s + 0.0579·35-s − 1.17·37-s + (0.811 − 1.40i)41-s + (0.283 + 0.491i)43-s + (−0.837 − 1.45i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(19.1166\)
Root analytic conductor: \(4.37225\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :3/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.582689783\)
\(L(\frac12)\) \(\approx\) \(1.582689783\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-1.5 + 2.59i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-2 - 3.46i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (12 + 20.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-12.5 + 21.6i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 21T + 4.91e3T^{2} \)
19 \( 1 + 52T + 6.85e3T^{2} \)
23 \( 1 + (-84 + 145. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (88.5 + 153. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-62 + 107. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 265T + 5.06e4T^{2} \)
41 \( 1 + (-213 + 368. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-80 - 138. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (270 + 467. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 258T + 1.48e5T^{2} \)
59 \( 1 + (-264 + 457. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-252.5 - 437. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-122 + 211. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 204T + 3.57e5T^{2} \)
73 \( 1 + 397T + 3.89e5T^{2} \)
79 \( 1 + (100 + 173. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (270 + 467. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 453T + 7.04e5T^{2} \)
97 \( 1 + (145 + 251. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.91132400799111995680600352012, −10.16882725731243315402947192665, −8.930750391776338823870777747760, −8.307836225339559491017752394858, −7.12244662999144441439645897627, −5.95158370162236395643904689288, −5.05446783411192447150563697961, −3.68809688264878242395316047953, −2.32430162655064692278530961642, −0.59717089334376277880800897747, 1.45698183046890134047804567472, 2.94698102145398600991324852252, 4.30083392447114756960241498679, 5.41373449529013079476264240796, 6.64664217196931679015027422098, 7.47208564725728379389780182553, 8.606695155713614427939266922939, 9.563417134852982954005889969649, 10.51980944338591987809947793868, 11.26768999843038118153524609986

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