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Properties

Label 2-18e2-4.3-c2-0-28
Degree $2$
Conductor $324$
Sign $0.159 + 0.987i$
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  + (−1.52 − 1.29i)2-s + (0.637 + 3.94i)4-s + 7.82·5-s − 12.3i·7-s + (4.14 − 6.83i)8-s + (−11.9 − 10.1i)10-s + 11.0i·11-s + 3.54·13-s + (−16.0 + 18.8i)14-s + (−15.1 + 5.03i)16-s + 8.77·17-s − 19.2i·19-s + (4.98 + 30.8i)20-s + (14.3 − 16.8i)22-s + 0.712i·23-s + ⋯
L(s)  = 1  + (−0.761 − 0.648i)2-s + (0.159 + 0.987i)4-s + 1.56·5-s − 1.76i·7-s + (0.518 − 0.854i)8-s + (−1.19 − 1.01i)10-s + 1.00i·11-s + 0.273·13-s + (−1.14 + 1.34i)14-s + (−0.949 + 0.314i)16-s + 0.516·17-s − 1.01i·19-s + (0.249 + 1.54i)20-s + (0.653 − 0.767i)22-s + 0.0309i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.159 + 0.987i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ 0.159 + 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.13389 - 0.965528i\)
\(L(\frac12)\) \(\approx\) \(1.13389 - 0.965528i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.52 + 1.29i)T \)
3 \( 1 \)
good5 \( 1 - 7.82T + 25T^{2} \)
7 \( 1 + 12.3iT - 49T^{2} \)
11 \( 1 - 11.0iT - 121T^{2} \)
13 \( 1 - 3.54T + 169T^{2} \)
17 \( 1 - 8.77T + 289T^{2} \)
19 \( 1 + 19.2iT - 361T^{2} \)
23 \( 1 - 0.712iT - 529T^{2} \)
29 \( 1 - 18.2T + 841T^{2} \)
31 \( 1 - 11.1iT - 961T^{2} \)
37 \( 1 + 35.5T + 1.36e3T^{2} \)
41 \( 1 + 12.1T + 1.68e3T^{2} \)
43 \( 1 + 66.2iT - 1.84e3T^{2} \)
47 \( 1 + 41.4iT - 2.20e3T^{2} \)
53 \( 1 - 74.7T + 2.80e3T^{2} \)
59 \( 1 + 20.7iT - 3.48e3T^{2} \)
61 \( 1 - 48.8T + 3.72e3T^{2} \)
67 \( 1 + 8.04iT - 4.48e3T^{2} \)
71 \( 1 - 87.9iT - 5.04e3T^{2} \)
73 \( 1 - 62.0T + 5.32e3T^{2} \)
79 \( 1 + 9.91iT - 6.24e3T^{2} \)
83 \( 1 + 22.1iT - 6.88e3T^{2} \)
89 \( 1 + 106.T + 7.92e3T^{2} \)
97 \( 1 - 131.T + 9.40e3T^{2} \)
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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

−10.71385874338212109317804261511, −10.23970832039302106186432156776, −9.661425415266785004260894661204, −8.608044298485401155305757049575, −7.23060286144055559533850340249, −6.74660112984741254259265454541, −5.02362942771037608131869757519, −3.73831253445245602034984708371, −2.19725926329778015278040343515, −1.00601922264511626470097903453, 1.55961979610414486353451626960, 2.72780206463235163396160758525, 5.30012257833654475243976677316, 5.83993254461452974660184746174, 6.43054634519610818402388194274, 8.132071256753392857958020655150, 8.822222905781088147887452243671, 9.559370237700225659302271454613, 10.32180910799249176330731386430, 11.43041809551309731662288303251

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