Properties

Degree $2$
Conductor $63$
Sign $-1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 7·4-s − 16·5-s − 7·7-s − 15·8-s − 16·10-s + 8·11-s + 28·13-s − 7·14-s + 41·16-s − 54·17-s − 110·19-s + 112·20-s + 8·22-s − 48·23-s + 131·25-s + 28·26-s + 49·28-s + 110·29-s + 12·31-s + 161·32-s − 54·34-s + 112·35-s − 246·37-s − 110·38-s + 240·40-s − 182·41-s + ⋯
L(s)  = 1  + 0.353·2-s − 7/8·4-s − 1.43·5-s − 0.377·7-s − 0.662·8-s − 0.505·10-s + 0.219·11-s + 0.597·13-s − 0.133·14-s + 0.640·16-s − 0.770·17-s − 1.32·19-s + 1.25·20-s + 0.0775·22-s − 0.435·23-s + 1.04·25-s + 0.211·26-s + 0.330·28-s + 0.704·29-s + 0.0695·31-s + 0.889·32-s − 0.272·34-s + 0.540·35-s − 1.09·37-s − 0.469·38-s + 0.948·40-s − 0.693·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-1$
Motivic weight: \(3\)
Character: $\chi_{63} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + p T \)
good2 \( 1 - T + p^{3} T^{2} \)
5 \( 1 + 16 T + p^{3} T^{2} \)
11 \( 1 - 8 T + p^{3} T^{2} \)
13 \( 1 - 28 T + p^{3} T^{2} \)
17 \( 1 + 54 T + p^{3} T^{2} \)
19 \( 1 + 110 T + p^{3} T^{2} \)
23 \( 1 + 48 T + p^{3} T^{2} \)
29 \( 1 - 110 T + p^{3} T^{2} \)
31 \( 1 - 12 T + p^{3} T^{2} \)
37 \( 1 + 246 T + p^{3} T^{2} \)
41 \( 1 + 182 T + p^{3} T^{2} \)
43 \( 1 - 128 T + p^{3} T^{2} \)
47 \( 1 + 324 T + p^{3} T^{2} \)
53 \( 1 - 162 T + p^{3} T^{2} \)
59 \( 1 + 810 T + p^{3} T^{2} \)
61 \( 1 + 8 p T + p^{3} T^{2} \)
67 \( 1 - 244 T + p^{3} T^{2} \)
71 \( 1 - 768 T + p^{3} T^{2} \)
73 \( 1 + 702 T + p^{3} T^{2} \)
79 \( 1 - 440 T + p^{3} T^{2} \)
83 \( 1 - 1302 T + p^{3} T^{2} \)
89 \( 1 + 730 T + p^{3} T^{2} \)
97 \( 1 - 294 T + p^{3} T^{2} \)
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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

−13.86173535710980529985891661879, −12.77453102155997503476856878985, −11.88459754933758645269265417528, −10.60063305698935451103233703308, −8.978615266778222122187380203428, −8.104649591618020347309056903192, −6.45332508856489766127488904836, −4.57819530593898402830050696633, −3.58703575599678934983580328821, 0, 3.58703575599678934983580328821, 4.57819530593898402830050696633, 6.45332508856489766127488904836, 8.104649591618020347309056903192, 8.978615266778222122187380203428, 10.60063305698935451103233703308, 11.88459754933758645269265417528, 12.77453102155997503476856878985, 13.86173535710980529985891661879

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