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Properties

Label 2-18e2-9.5-c2-0-5
Degree $2$
Conductor $324$
Sign $0.642 + 0.766i$
Analytic cond. $8.82836$
Root an. cond. $2.97125$
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 − 1.73i)7-s + (11 − 19.0i)13-s + 26·19-s + (−12.5 − 21.6i)25-s + (23 − 39.8i)31-s + 26·37-s + (11 + 19.0i)43-s + (22.5 − 38.9i)49-s + (−37 − 64.0i)61-s + (−61 + 105. i)67-s − 46·73-s + (71 + 122. i)79-s − 44·91-s + (−1 − 1.73i)97-s + (−97 + 168. i)103-s + ⋯
L(s)  = 1  + (−0.142 − 0.247i)7-s + (0.846 − 1.46i)13-s + 1.36·19-s + (−0.5 − 0.866i)25-s + (0.741 − 1.28i)31-s + 0.702·37-s + (0.255 + 0.443i)43-s + (0.459 − 0.795i)49-s + (−0.606 − 1.05i)61-s + (−0.910 + 1.57i)67-s − 0.630·73-s + (0.898 + 1.55i)79-s − 0.483·91-s + (−0.0103 − 0.0178i)97-s + (−0.941 + 1.63i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.45398 - 0.678005i\)
\(L(\frac12)\) \(\approx\) \(1.45398 - 0.678005i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-11 + 19.0i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 26T + 361T^{2} \)
23 \( 1 + (264.5 + 458. i)T^{2} \)
29 \( 1 + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-23 + 39.8i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 26T + 1.36e3T^{2} \)
41 \( 1 + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-11 - 19.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (37 + 64.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (61 - 105. i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 46T + 5.32e3T^{2} \)
79 \( 1 + (-71 - 122. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.24870495957802372274806804232, −10.29897743446767198834776606588, −9.545834382620807189776801707870, −8.271583470334592508435732280786, −7.60213509009136175900295188382, −6.27532358881692826432371743201, −5.39397395110852450372549732774, −3.98590699455766103659369546543, −2.81002127754550405051241648078, −0.857693626731538339133446307241, 1.48974581908860858299831023682, 3.15060417066673122372829858447, 4.38872522973272717743882461815, 5.65133508923564473862139800688, 6.66902607770674712175417211026, 7.66784893863879694413708288831, 8.901879704569014248514364090232, 9.480135932541500706675560709822, 10.67807634434961173421128829986, 11.60964300009599921997408217136

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