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  − 5-s + 5·11-s + 4·17-s − 8·19-s − 4·23-s − 4·25-s + 5·29-s − 3·31-s − 4·37-s − 2·43-s − 6·47-s + 9·53-s − 5·55-s − 11·59-s − 6·61-s + 2·67-s + 2·71-s + 10·73-s − 3·79-s − 7·83-s − 4·85-s + 6·89-s + 8·95-s + 7·97-s − 10·101-s − 8·103-s + 3·107-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.50·11-s + 0.970·17-s − 1.83·19-s − 0.834·23-s − 4/5·25-s + 0.928·29-s − 0.538·31-s − 0.657·37-s − 0.304·43-s − 0.875·47-s + 1.23·53-s − 0.674·55-s − 1.43·59-s − 0.768·61-s + 0.244·67-s + 0.237·71-s + 1.17·73-s − 0.337·79-s − 0.768·83-s − 0.433·85-s + 0.635·89-s + 0.820·95-s + 0.710·97-s − 0.995·101-s − 0.788·103-s + 0.290·107-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 \( 1 + T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−7.68309275325874930767113524989, −6.74718927901669286156292621356, −6.34193348688004965261820540292, −5.56841296393293083143396700910, −4.52793828627858797400311200875, −3.97236916296227689998674910498, −3.36467853734859365806748842623, −2.16186839989639863953108016125, −1.32908498934439305192392897250, 0, 1.32908498934439305192392897250, 2.16186839989639863953108016125, 3.36467853734859365806748842623, 3.97236916296227689998674910498, 4.52793828627858797400311200875, 5.56841296393293083143396700910, 6.34193348688004965261820540292, 6.74718927901669286156292621356, 7.68309275325874930767113524989

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