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Properties

Label 2-18e2-4.3-c2-0-35
Degree $2$
Conductor $324$
Sign $0.949 + 0.315i$
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  + (1.97 + 0.319i)2-s + (3.79 + 1.26i)4-s + 2.71·5-s − 11.5i·7-s + (7.09 + 3.70i)8-s + (5.35 + 0.865i)10-s − 9.87i·11-s − 0.592·13-s + (3.70 − 22.8i)14-s + (12.8 + 9.57i)16-s + 8.87·17-s + 14.0i·19-s + (10.2 + 3.41i)20-s + (3.15 − 19.4i)22-s + 21.1i·23-s + ⋯
L(s)  = 1  + (0.987 + 0.159i)2-s + (0.949 + 0.315i)4-s + 0.542·5-s − 1.65i·7-s + (0.886 + 0.462i)8-s + (0.535 + 0.0865i)10-s − 0.897i·11-s − 0.0455·13-s + (0.264 − 1.63i)14-s + (0.801 + 0.598i)16-s + 0.522·17-s + 0.742i·19-s + (0.514 + 0.170i)20-s + (0.143 − 0.885i)22-s + 0.917i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.315i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.949 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.949 + 0.315i$
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: \((2,\ 324,\ (\ :1),\ 0.949 + 0.315i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.31195 - 0.535501i\)
\(L(\frac12)\) \(\approx\) \(3.31195 - 0.535501i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.97 - 0.319i)T \)
3 \( 1 \)
good5 \( 1 - 2.71T + 25T^{2} \)
7 \( 1 + 11.5iT - 49T^{2} \)
11 \( 1 + 9.87iT - 121T^{2} \)
13 \( 1 + 0.592T + 169T^{2} \)
17 \( 1 - 8.87T + 289T^{2} \)
19 \( 1 - 14.0iT - 361T^{2} \)
23 \( 1 - 21.1iT - 529T^{2} \)
29 \( 1 - 20.3T + 841T^{2} \)
31 \( 1 - 16.5iT - 961T^{2} \)
37 \( 1 + 40.6T + 1.36e3T^{2} \)
41 \( 1 - 42.4T + 1.68e3T^{2} \)
43 \( 1 - 37.2iT - 1.84e3T^{2} \)
47 \( 1 - 1.81iT - 2.20e3T^{2} \)
53 \( 1 - 21.1T + 2.80e3T^{2} \)
59 \( 1 + 88.5iT - 3.48e3T^{2} \)
61 \( 1 + 72.9T + 3.72e3T^{2} \)
67 \( 1 + 44.2iT - 4.48e3T^{2} \)
71 \( 1 - 111. iT - 5.04e3T^{2} \)
73 \( 1 + 76.2T + 5.32e3T^{2} \)
79 \( 1 - 9.58iT - 6.24e3T^{2} \)
83 \( 1 - 85.0iT - 6.88e3T^{2} \)
89 \( 1 + 64.7T + 7.92e3T^{2} \)
97 \( 1 - 7.18T + 9.40e3T^{2} \)
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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

−11.35134298775985114604815367417, −10.58295914036349838044604045664, −9.802781314052766376912919052016, −8.156183598349048457199559915263, −7.34236265373267864034122139556, −6.33494809248553599340297009897, −5.39165046672139487090599709265, −4.11784224661275057717107589310, −3.23301843464565841611601630529, −1.38364283666699502175746159955, 1.97785574485238635619839986453, 2.82719278528321257212739699844, 4.49634792630581704818272149824, 5.48787069147499767573014934443, 6.20778028168170897345046394593, 7.35949438556831968214135033482, 8.731931305592327280468301127509, 9.701369058374985344767466805065, 10.62223463251800638532535754286, 11.90513699404079925706699320030

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