Properties

Degree $2$
Conductor $7056$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 2·11-s + 4·13-s + 6·17-s + 8·19-s − 6·23-s − 25-s + 10·29-s + 4·31-s + 6·37-s − 6·41-s − 4·43-s − 8·47-s − 2·53-s + 4·55-s + 4·59-s + 8·61-s + 8·65-s + 8·67-s − 10·71-s − 4·73-s − 4·79-s − 12·83-s + 12·85-s − 14·89-s + 16·95-s − 4·97-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.603·11-s + 1.10·13-s + 1.45·17-s + 1.83·19-s − 1.25·23-s − 1/5·25-s + 1.85·29-s + 0.718·31-s + 0.986·37-s − 0.937·41-s − 0.609·43-s − 1.16·47-s − 0.274·53-s + 0.539·55-s + 0.520·59-s + 1.02·61-s + 0.992·65-s + 0.977·67-s − 1.18·71-s − 0.468·73-s − 0.450·79-s − 1.31·83-s + 1.30·85-s − 1.48·89-s + 1.64·95-s − 0.406·97-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: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.193789630\)
\(L(\frac12)\) \(\approx\) \(3.193789630\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−8.214049555678150897243332023210, −7.15754182145600740052829504413, −6.42545743232601162444824595306, −5.83278720674609880318900252712, −5.32599066553471681281248421413, −4.33905542581195468011822208863, −3.45541409663235621019400347394, −2.83220699652796951187619347471, −1.57197913673115120028316812944, −1.03861269488820815842620666887, 1.03861269488820815842620666887, 1.57197913673115120028316812944, 2.83220699652796951187619347471, 3.45541409663235621019400347394, 4.33905542581195468011822208863, 5.32599066553471681281248421413, 5.83278720674609880318900252712, 6.42545743232601162444824595306, 7.15754182145600740052829504413, 8.214049555678150897243332023210

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