Properties

Degree $2$
Conductor $7056$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s − 3·11-s − 2·13-s + 6·17-s + 2·19-s − 6·23-s + 4·25-s − 9·29-s − 7·31-s − 10·37-s + 4·43-s − 12·47-s + 3·53-s − 9·55-s + 3·59-s + 4·61-s − 6·65-s − 2·67-s − 2·73-s − 5·79-s − 9·83-s + 18·85-s − 6·89-s + 6·95-s + 13·97-s + 6·101-s − 16·103-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.904·11-s − 0.554·13-s + 1.45·17-s + 0.458·19-s − 1.25·23-s + 4/5·25-s − 1.67·29-s − 1.25·31-s − 1.64·37-s + 0.609·43-s − 1.75·47-s + 0.412·53-s − 1.21·55-s + 0.390·59-s + 0.512·61-s − 0.744·65-s − 0.244·67-s − 0.234·73-s − 0.562·79-s − 0.987·83-s + 1.95·85-s − 0.635·89-s + 0.615·95-s + 1.31·97-s + 0.597·101-s − 1.57·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{7056} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−7.55897061209615858126116845368, −6.97910161712974459656407065808, −5.90823736302778728203678186086, −5.52219615915251146899126346033, −5.09851582433884081701270571273, −3.85477814317456269292247844951, −3.08421331729666561174533371760, −2.12501224527283536650390996746, −1.55872607484065073233825464189, 0, 1.55872607484065073233825464189, 2.12501224527283536650390996746, 3.08421331729666561174533371760, 3.85477814317456269292247844951, 5.09851582433884081701270571273, 5.52219615915251146899126346033, 5.90823736302778728203678186086, 6.97910161712974459656407065808, 7.55897061209615858126116845368

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