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Properties

Label 2-18e2-36.31-c2-0-25
Degree $2$
Conductor $324$
Sign $0.566 - 0.824i$
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  + (0.270 + 1.98i)2-s + (−3.85 + 1.07i)4-s + (0.540 − 0.935i)5-s + (5.20 − 3.00i)7-s + (−3.16 − 7.34i)8-s + (2 + 0.817i)10-s + (15.3 − 8.86i)11-s + (−6.20 + 10.7i)13-s + (7.36 + 9.50i)14-s + (13.7 − 8.25i)16-s + 26.3·17-s + 5.19i·19-s + (−1.08 + 4.18i)20-s + (21.7 + 28.0i)22-s + (−25.8 − 14.9i)23-s + ⋯
L(s)  = 1  + (0.135 + 0.990i)2-s + (−0.963 + 0.267i)4-s + (0.108 − 0.187i)5-s + (0.744 − 0.429i)7-s + (−0.395 − 0.918i)8-s + (0.200 + 0.0817i)10-s + (1.39 − 0.805i)11-s + (−0.477 + 0.827i)13-s + (0.526 + 0.679i)14-s + (0.856 − 0.515i)16-s + 1.55·17-s + 0.273i·19-s + (−0.0540 + 0.209i)20-s + (0.986 + 1.27i)22-s + (−1.12 − 0.648i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.566 - 0.824i$
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: \((2,\ 324,\ (\ :1),\ 0.566 - 0.824i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.64858 + 0.867775i\)
\(L(\frac12)\) \(\approx\) \(1.64858 + 0.867775i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.270 - 1.98i)T \)
3 \( 1 \)
good5 \( 1 + (-0.540 + 0.935i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-5.20 + 3.00i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-15.3 + 8.86i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (6.20 - 10.7i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 26.3T + 289T^{2} \)
19 \( 1 - 5.19iT - 361T^{2} \)
23 \( 1 + (25.8 + 14.9i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-2.16 - 3.74i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-38.8 - 22.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 20.4T + 1.36e3T^{2} \)
41 \( 1 + (-29.6 + 51.3i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-16.5 + 9.57i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (35.5 - 20.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 70.0T + 2.80e3T^{2} \)
59 \( 1 + (-25.0 - 14.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (9.20 + 15.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (82.1 + 47.4i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 83.8iT - 5.04e3T^{2} \)
73 \( 1 - 55.8T + 5.32e3T^{2} \)
79 \( 1 + (35.5 - 20.5i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (30.7 - 17.7i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 26.3T + 7.92e3T^{2} \)
97 \( 1 + (38.1 + 66.1i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.82971162619152510905465725784, −10.45515008157216412674558693435, −9.401032166172701624605617171327, −8.583559823694672800208066260978, −7.67604498149457677866748374066, −6.68860465092559489022046626322, −5.71438190528438535391203348189, −4.57832482960256113093783431757, −3.60308213802713160886740464618, −1.17627217650322922865146879573, 1.24723021255041196314414304748, 2.57968620643799072133218407307, 3.95235774545545307016120264434, 5.01907998916287142716307545246, 6.09676646255278323557699101862, 7.66782890879897403242915804180, 8.566904155443949167103278024319, 9.800348738749407605817330379646, 10.13741557694851504586283449109, 11.59727427171209491013073094486

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