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 + 6·13-s − 5·17-s + 19-s − 7·23-s − 4·25-s − 2·29-s − 5·31-s + 3·37-s − 2·41-s + 4·43-s − 5·47-s + 53-s − 3·55-s − 15·59-s + 5·61-s − 6·65-s + 9·67-s − 7·73-s − 79-s − 12·83-s + 5·85-s + 7·89-s − 95-s + 2·97-s + 3·101-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.904·11-s + 1.66·13-s − 1.21·17-s + 0.229·19-s − 1.45·23-s − 4/5·25-s − 0.371·29-s − 0.898·31-s + 0.493·37-s − 0.312·41-s + 0.609·43-s − 0.729·47-s + 0.137·53-s − 0.404·55-s − 1.95·59-s + 0.640·61-s − 0.744·65-s + 1.09·67-s − 0.819·73-s − 0.112·79-s − 1.31·83-s + 0.542·85-s + 0.741·89-s − 0.102·95-s + 0.203·97-s + 0.298·101-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 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
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

−7.68028487353977020685434420935, −6.81299387469983938345337002747, −6.18283629506126241248271512806, −5.69553024467688056130746201791, −4.48722885144786324382387208225, −3.93920665375416635707988839718, −3.41923492434141787529341330204, −2.13434730284385344198940692364, −1.34283087210270614855046078004, 0, 1.34283087210270614855046078004, 2.13434730284385344198940692364, 3.41923492434141787529341330204, 3.93920665375416635707988839718, 4.48722885144786324382387208225, 5.69553024467688056130746201791, 6.18283629506126241248271512806, 6.81299387469983938345337002747, 7.68028487353977020685434420935

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