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  − 4·2-s + 8·4-s + 4·5-s − 7·7-s − 16·10-s − 62·11-s − 62·13-s + 28·14-s − 64·16-s − 84·17-s + 100·19-s + 32·20-s + 248·22-s + 42·23-s − 109·25-s + 248·26-s − 56·28-s + 10·29-s − 48·31-s + 256·32-s + 336·34-s − 28·35-s − 246·37-s − 400·38-s + 248·41-s + 68·43-s − 496·44-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.357·5-s − 0.377·7-s − 0.505·10-s − 1.69·11-s − 1.32·13-s + 0.534·14-s − 16-s − 1.19·17-s + 1.20·19-s + 0.357·20-s + 2.40·22-s + 0.380·23-s − 0.871·25-s + 1.87·26-s − 0.377·28-s + 0.0640·29-s − 0.278·31-s + 1.41·32-s + 1.69·34-s − 0.135·35-s − 1.09·37-s − 1.70·38-s + 0.944·41-s + 0.241·43-s − 1.69·44-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 \( 1 + p^{2} T + p^{3} T^{2} \)
5 \( 1 - 4 T + p^{3} T^{2} \)
11 \( 1 + 62 T + p^{3} T^{2} \)
13 \( 1 + 62 T + p^{3} T^{2} \)
17 \( 1 + 84 T + p^{3} T^{2} \)
19 \( 1 - 100 T + p^{3} T^{2} \)
23 \( 1 - 42 T + p^{3} T^{2} \)
29 \( 1 - 10 T + p^{3} T^{2} \)
31 \( 1 + 48 T + p^{3} T^{2} \)
37 \( 1 + 246 T + p^{3} T^{2} \)
41 \( 1 - 248 T + p^{3} T^{2} \)
43 \( 1 - 68 T + p^{3} T^{2} \)
47 \( 1 + 324 T + p^{3} T^{2} \)
53 \( 1 + 258 T + p^{3} T^{2} \)
59 \( 1 + 120 T + p^{3} T^{2} \)
61 \( 1 - 622 T + p^{3} T^{2} \)
67 \( 1 - 904 T + p^{3} T^{2} \)
71 \( 1 - 678 T + p^{3} T^{2} \)
73 \( 1 + 642 T + p^{3} T^{2} \)
79 \( 1 - 740 T + p^{3} T^{2} \)
83 \( 1 + 468 T + p^{3} T^{2} \)
89 \( 1 + 200 T + p^{3} T^{2} \)
97 \( 1 + 1266 T + p^{3} T^{2} \)
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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

−13.82589962527974153383578973793, −12.77865957707603076496688394900, −11.19822933075786614988986465109, −10.10951440717806364602568485162, −9.408822223293877112685034241746, −8.028922084232416107968359880777, −7.05741833233961856532076399114, −5.14093606095801719029693580969, −2.39823772414700851618467437770, 0, 2.39823772414700851618467437770, 5.14093606095801719029693580969, 7.05741833233961856532076399114, 8.028922084232416107968359880777, 9.408822223293877112685034241746, 10.10951440717806364602568485162, 11.19822933075786614988986465109, 12.77865957707603076496688394900, 13.82589962527974153383578973793

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