Properties

Label 2-18e2-36.31-c2-0-3
Degree $2$
Conductor $324$
Sign $-0.173 - 0.984i$
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)2-s + (−1.99 − 3.46i)4-s + (−1 + 1.73i)5-s + (−6 + 3.46i)7-s − 7.99·8-s + (1.99 + 3.46i)10-s + (−6 + 3.46i)11-s + (−1 + 1.73i)13-s + 13.8i·14-s + (−8 + 13.8i)16-s − 10·17-s + 20.7i·19-s + 7.99·20-s + 13.8i·22-s + (−24 − 13.8i)23-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.200 + 0.346i)5-s + (−0.857 + 0.494i)7-s − 0.999·8-s + (0.199 + 0.346i)10-s + (−0.545 + 0.314i)11-s + (−0.0769 + 0.133i)13-s + 0.989i·14-s + (−0.5 + 0.866i)16-s − 0.588·17-s + 1.09i·19-s + 0.399·20-s + 0.629i·22-s + (−1.04 − 0.602i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ -0.173 - 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.221450 + 0.263914i\)
\(L(\frac12)\) \(\approx\) \(0.221450 + 0.263914i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 \)
good5 \( 1 + (1 - 1.73i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (6 - 3.46i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (6 - 3.46i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 10T + 289T^{2} \)
19 \( 1 - 20.7iT - 361T^{2} \)
23 \( 1 + (24 + 13.8i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (13 + 22.5i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (6 + 3.46i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 26T + 1.36e3T^{2} \)
41 \( 1 + (-29 + 50.2i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (42 - 24.2i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (60 - 34.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 74T + 2.80e3T^{2} \)
59 \( 1 + (78 + 45.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (13 + 22.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (6 + 3.46i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 46T + 5.32e3T^{2} \)
79 \( 1 + (-102 + 58.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (42 - 24.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 82T + 7.92e3T^{2} \)
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

−11.72961434489786157942847585850, −10.78402659816629405112350240089, −9.940984964279912904362777562420, −9.205668706343231434137846308469, −7.940534805112929077970162727799, −6.51150779221721557336402637020, −5.67412141302988541619372049614, −4.34647083206120352549128820763, −3.22786289856363845548582311928, −2.09493291240081836587441258874, 0.12925517965759771049272670754, 2.92711068849275110330179030511, 4.10245543068972086000529079829, 5.15759336281776685108476172514, 6.30318045579560529415228275301, 7.14482798578321806455600881390, 8.145003103636377732392346662195, 9.053910286263710746606062638145, 10.06030159042584875631355785039, 11.29387342887454806574607139596

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