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Properties

Label 2-18e2-9.5-c2-0-0
Degree $2$
Conductor $324$
Sign $-0.819 - 0.573i$
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  + (−1.34 − 0.776i)5-s + (−2.09 − 3.63i)7-s + (−13.7 + 7.91i)11-s + (−4.69 + 8.13i)13-s + 23.9i·17-s + 18.9·19-s + (−21.7 − 12.5i)23-s + (−11.2 − 19.5i)25-s + (−47.8 + 27.6i)29-s + (−19.5 + 33.9i)31-s + 6.51i·35-s + 19.1·37-s + (−2.42 − 1.40i)41-s + (−16.8 − 29.2i)43-s + (−13.9 + 8.06i)47-s + ⋯
L(s)  = 1  + (−0.268 − 0.155i)5-s + (−0.299 − 0.519i)7-s + (−1.24 + 0.719i)11-s + (−0.361 + 0.625i)13-s + 1.40i·17-s + 0.998·19-s + (−0.947 − 0.546i)23-s + (−0.451 − 0.782i)25-s + (−1.65 + 0.952i)29-s + (−0.631 + 1.09i)31-s + 0.186i·35-s + 0.518·37-s + (−0.0591 − 0.0341i)41-s + (−0.392 − 0.680i)43-s + (−0.297 + 0.171i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.819 - 0.573i$
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: \((2,\ 324,\ (\ :1),\ -0.819 - 0.573i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.121038 + 0.383883i\)
\(L(\frac12)\) \(\approx\) \(0.121038 + 0.383883i\)
\(L(2)\) 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 \( 1 + (1.34 + 0.776i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (2.09 + 3.63i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (13.7 - 7.91i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (4.69 - 8.13i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 23.9iT - 289T^{2} \)
19 \( 1 - 18.9T + 361T^{2} \)
23 \( 1 + (21.7 + 12.5i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (47.8 - 27.6i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (19.5 - 33.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 19.1T + 1.36e3T^{2} \)
41 \( 1 + (2.42 + 1.40i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (16.8 + 29.2i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (13.9 - 8.06i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 53.7iT - 2.80e3T^{2} \)
59 \( 1 + (8.59 + 4.96i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-31.3 - 54.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (10.2 - 17.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 113. iT - 5.04e3T^{2} \)
73 \( 1 + 110.T + 5.32e3T^{2} \)
79 \( 1 + (20.3 + 35.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-142. + 82.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 13.6iT - 7.92e3T^{2} \)
97 \( 1 + (-77.1 - 133. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.89273669889849034028099470548, −10.61355876943625030195649539363, −10.11930870825682574859709432042, −8.965093184041040805640950524769, −7.83147954090071906048049214546, −7.15200827017597161531207005424, −5.84925014134259473037730857921, −4.67626562594540861416013220431, −3.57772036723962910700712030798, −1.95099200821871735811686655464, 0.17724233507019846450033173011, 2.49029444710459692127393260877, 3.53673459395072288819965077763, 5.23311678051800818941535895060, 5.84254820724505735106043190146, 7.47927637667429934033485953413, 7.899915162461611173835620037848, 9.343582145373317004445986111919, 9.930595703594355803830222052579, 11.27538486110882970800221366530

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