Properties

Label 2-18e2-9.4-c5-0-16
Degree $2$
Conductor $324$
Sign $0.173 + 0.984i$
Analytic cond. $51.9643$
Root an. cond. $7.20863$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (44.0 − 76.3i)5-s + (−14.5 − 25.1i)7-s + (44.0 + 76.3i)11-s + (−164.5 + 284. i)13-s + 2.20e3·17-s + 1.79e3·19-s + (1.80e3 − 3.13e3i)23-s + (−2.32e3 − 4.02e3i)25-s + (705. + 1.22e3i)29-s + (−2.61e3 + 4.52e3i)31-s − 2.55e3·35-s + 8.78e3·37-s + (−7.75e3 + 1.34e4i)41-s + (−9.98e3 − 1.72e4i)43-s + (−5.42e3 − 9.39e3i)47-s + ⋯
L(s)  = 1  + (0.788 − 1.36i)5-s + (−0.111 − 0.193i)7-s + (0.109 + 0.190i)11-s + (−0.269 + 0.467i)13-s + 1.85·17-s + 1.14·19-s + (0.712 − 1.23i)23-s + (−0.744 − 1.28i)25-s + (0.155 + 0.269i)29-s + (−0.488 + 0.846i)31-s − 0.352·35-s + 1.05·37-s + (−0.720 + 1.24i)41-s + (−0.823 − 1.42i)43-s + (−0.358 − 0.620i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(51.9643\)
Root analytic conductor: \(7.20863\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :5/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.542562658\)
\(L(\frac12)\) \(\approx\) \(2.542562658\)
\(L(\frac{7}{2})\) 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 \( 1 + (-44.0 + 76.3i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (14.5 + 25.1i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-44.0 - 76.3i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (164.5 - 284. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 - 2.20e3T + 1.41e6T^{2} \)
19 \( 1 - 1.79e3T + 2.47e6T^{2} \)
23 \( 1 + (-1.80e3 + 3.13e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (-705. - 1.22e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (2.61e3 - 4.52e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 8.78e3T + 6.93e7T^{2} \)
41 \( 1 + (7.75e3 - 1.34e4i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (9.98e3 + 1.72e4i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (5.42e3 + 9.39e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + 2.94e4T + 4.18e8T^{2} \)
59 \( 1 + (-2.86e3 + 4.96e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-534.5 - 925. i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-3.10e4 + 5.37e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 4.63e4T + 1.80e9T^{2} \)
73 \( 1 + 4.80e4T + 2.07e9T^{2} \)
79 \( 1 + (2.49e4 + 4.32e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (2.88e4 + 4.99e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 - 8.79e4T + 5.58e9T^{2} \)
97 \( 1 + (6.45e3 + 1.11e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.27888087487137627070717247499, −9.628179708349334470055267108466, −8.824331437213691136359019818100, −7.83040857884183382887616468972, −6.62572848272687237131323087390, −5.37816597507941014548943976947, −4.81338017822091862590186409125, −3.29709568770489529308794944179, −1.67167902106029621960973552367, −0.74705358850721269413874412768, 1.25546665282057603361865804219, 2.76353421511357337076510849333, 3.43716423012111872291720952693, 5.36413274222223989383758186028, 6.00874814335325742718141907941, 7.18990126672394278072781452705, 7.891464447079474372680076104793, 9.608450411023734705203120072542, 9.833323078398936136067122093507, 10.97343374124205623267800613667

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