Properties

Degree $2$
Conductor $163254$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 2·5-s + 6-s − 7-s − 8-s + 9-s − 2·10-s + 4·11-s − 12-s + 14-s − 2·15-s + 16-s − 6·17-s − 18-s − 4·19-s + 2·20-s + 21-s − 4·22-s − 23-s + 24-s − 25-s − 27-s − 28-s − 2·29-s + 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.288·12-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.218·21-s − 0.852·22-s − 0.208·23-s + 0.204·24-s − 1/5·25-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(163254\)    =    \(2 \cdot 3 \cdot 7 \cdot 13^{2} \cdot 23\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{163254} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 163254,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 \)
23 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.52935188132935, −13.00014249451384, −12.41344512942106, −12.00841213637971, −11.60825323699230, −10.87702185701595, −10.74734079937815, −10.08308492450342, −9.721213247187618, −9.106974467899618, −8.934265028358095, −8.358067540696352, −7.649108491797431, −7.027072003377404, −6.571955517078663, −6.233967830205165, −5.941350783971278, −5.174352323363938, −4.490886974829638, −4.065708950190296, −3.398111237879318, −2.470238362517657, −2.111483879994781, −1.498070697826239, −0.7744694638098056, 0, 0.7744694638098056, 1.498070697826239, 2.111483879994781, 2.470238362517657, 3.398111237879318, 4.065708950190296, 4.490886974829638, 5.174352323363938, 5.941350783971278, 6.233967830205165, 6.571955517078663, 7.027072003377404, 7.649108491797431, 8.358067540696352, 8.934265028358095, 9.106974467899618, 9.721213247187618, 10.08308492450342, 10.74734079937815, 10.87702185701595, 11.60825323699230, 12.00841213637971, 12.41344512942106, 13.00014249451384, 13.52935188132935

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