Properties

Label 2-18e2-9.7-c5-0-0
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

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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: $\chi_{324} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :5/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.4744947847\)
\(L(\frac12)\) \(\approx\) \(0.4744947847\)
\(L(\frac{7}{2})\) 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 \( 1 + (6.45e3 - 1.11e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−11.22407057484587034877800668733, −9.899894912493468821325368237508, −8.982762319870874236949850286753, −8.257537963660596713387573196138, −7.34669652043767713883976965375, −5.99093022276243437509658538085, −4.80054777104255867474469001166, −4.14530909256683027975470334511, −2.51855257169075631848876412270, −0.927100071519429410835367512197, 0.15194050770754977453500940621, 2.15074753552206696132617107140, 3.33989667236541766142447681075, 4.24242363037715205516459641607, 5.76582560718863177388792798310, 7.03069124297558231810007965495, 7.34382004979477732296104575439, 8.661506630627305407320923916818, 9.751984196054516759630984586214, 10.76773255741567601527259090928

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