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Properties

Label 2-18e2-36.7-c2-0-44
Degree $2$
Conductor $324$
Sign $-0.766 - 0.642i$
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 + (−4 − 6.92i)5-s − 7.99·8-s − 15.9·10-s + (5 + 8.66i)13-s + (−8 + 13.8i)16-s − 16·17-s + (−15.9 + 27.7i)20-s + (−19.4 + 33.7i)25-s + 20·26-s + (20 − 34.6i)29-s + (15.9 + 27.7i)32-s + (−16 + 27.7i)34-s − 70·37-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.800 − 1.38i)5-s − 0.999·8-s − 1.59·10-s + (0.384 + 0.666i)13-s + (−0.5 + 0.866i)16-s − 0.941·17-s + (−0.799 + 1.38i)20-s + (−0.779 + 1.35i)25-s + 0.769·26-s + (0.689 − 1.19i)29-s + (0.499 + 0.866i)32-s + (−0.470 + 0.815i)34-s − 1.89·37-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ -0.766 - 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.327742 + 0.900466i\)
\(L(\frac12)\) \(\approx\) \(0.327742 + 0.900466i\)
\(L(2)\) 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 \( 1 + (4 + 6.92i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-5 - 8.66i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 16T + 289T^{2} \)
19 \( 1 - 361T^{2} \)
23 \( 1 + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-20 + 34.6i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (480.5 - 832. i)T^{2} \)
37 \( 1 + 70T + 1.36e3T^{2} \)
41 \( 1 + (40 + 69.2i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 56T + 2.80e3T^{2} \)
59 \( 1 + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-11 + 19.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 110T + 5.32e3T^{2} \)
79 \( 1 + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 160T + 7.92e3T^{2} \)
97 \( 1 + (-65 + 112. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.11829084628877236405602404271, −9.985312261069355399978262221877, −8.870837503566441489089058178462, −8.444717379838500010018585427938, −6.81358686711262502366751244350, −5.41921783526817406563375777299, −4.50801195833676593463465174585, −3.71672629168417051970807388874, −1.89116904659612649578775882992, −0.37919183467176667099697283802, 2.90429620806351092847395531762, 3.75164624348903047542013367409, 5.03819507206687820226401091361, 6.41472933347097982395915889757, 6.97973803770808077768758938712, 7.947767147872384359725612164932, 8.794368793698016836283302189183, 10.26350244793926142092367740105, 11.12027530804224371655854997908, 11.98636428030059413797197522445

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