Properties

Degree $2$
Conductor $6272$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·5-s + 9-s − 2·11-s + 2·13-s − 4·15-s + 2·17-s − 2·19-s − 4·23-s − 25-s + 4·27-s + 6·29-s + 4·33-s − 10·37-s − 4·39-s + 6·41-s + 6·43-s + 2·45-s − 8·47-s − 4·51-s + 6·53-s − 4·55-s + 4·57-s − 14·59-s + 2·61-s + 4·65-s + 10·67-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 1.03·15-s + 0.485·17-s − 0.458·19-s − 0.834·23-s − 1/5·25-s + 0.769·27-s + 1.11·29-s + 0.696·33-s − 1.64·37-s − 0.640·39-s + 0.937·41-s + 0.914·43-s + 0.298·45-s − 1.16·47-s − 0.560·51-s + 0.824·53-s − 0.539·55-s + 0.529·57-s − 1.82·59-s + 0.256·61-s + 0.496·65-s + 1.22·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6272\)    =    \(2^{7} \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{6272} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6272,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−7.64854981133969450065979207065, −6.69572961025283119298928810452, −6.12267427106713469054639441773, −5.67353733917564066154228887308, −5.04180686058605300396688448066, −4.23288044988283905382688289193, −3.14728661020841746026216746182, −2.19008846182011537870041620099, −1.21203684908075352649921343195, 0, 1.21203684908075352649921343195, 2.19008846182011537870041620099, 3.14728661020841746026216746182, 4.23288044988283905382688289193, 5.04180686058605300396688448066, 5.67353733917564066154228887308, 6.12267427106713469054639441773, 6.69572961025283119298928810452, 7.64854981133969450065979207065

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