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 + 3·11-s − 4·13-s − 4·19-s + 8·23-s − 4·25-s + 3·29-s − 5·31-s + 8·37-s + 8·41-s − 6·43-s − 10·47-s − 9·53-s − 3·55-s + 5·59-s + 10·61-s + 4·65-s − 6·67-s + 10·71-s − 2·73-s − 11·79-s − 7·83-s − 18·89-s + 4·95-s + 17·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.904·11-s − 1.10·13-s − 0.917·19-s + 1.66·23-s − 4/5·25-s + 0.557·29-s − 0.898·31-s + 1.31·37-s + 1.24·41-s − 0.914·43-s − 1.45·47-s − 1.23·53-s − 0.404·55-s + 0.650·59-s + 1.28·61-s + 0.496·65-s − 0.733·67-s + 1.18·71-s − 0.234·73-s − 1.23·79-s − 0.768·83-s − 1.90·89-s + 0.410·95-s + 1.72·97-s + 0.0995·101-s + 0.0985·103-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: $\chi_{7056} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
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 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−17.37669901729594, −16.90973796748329, −16.39557729494412, −15.68656931473645, −14.92631198449814, −14.66967646142017, −14.14012434189060, −13.03320338816441, −12.85623561985752, −12.09412993508959, −11.29397333894987, −11.18892020024520, −10.08796346463210, −9.594071792114660, −8.954917517785920, −8.251804295526352, −7.560769088533562, −6.899420018074602, −6.349933891838699, −5.407998281562053, −4.638061335393176, −4.062547500693504, −3.160441776820462, −2.321962408831111, −1.250430438185986, 0, 1.250430438185986, 2.321962408831111, 3.160441776820462, 4.062547500693504, 4.638061335393176, 5.407998281562053, 6.349933891838699, 6.899420018074602, 7.560769088533562, 8.251804295526352, 8.954917517785920, 9.594071792114660, 10.08796346463210, 11.18892020024520, 11.29397333894987, 12.09412993508959, 12.85623561985752, 13.03320338816441, 14.14012434189060, 14.66967646142017, 14.92631198449814, 15.68656931473645, 16.39557729494412, 16.90973796748329, 17.37669901729594

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