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  + 5-s − 5·11-s − 2·13-s − 6·17-s + 2·19-s + 6·23-s − 4·25-s − 3·29-s + 5·31-s − 2·37-s − 8·41-s + 4·43-s + 4·47-s + 9·53-s − 5·55-s − 3·59-s + 12·61-s − 2·65-s − 2·67-s + 8·71-s + 14·73-s − 79-s + 17·83-s − 6·85-s − 18·89-s + 2·95-s − 3·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.50·11-s − 0.554·13-s − 1.45·17-s + 0.458·19-s + 1.25·23-s − 4/5·25-s − 0.557·29-s + 0.898·31-s − 0.328·37-s − 1.24·41-s + 0.609·43-s + 0.583·47-s + 1.23·53-s − 0.674·55-s − 0.390·59-s + 1.53·61-s − 0.248·65-s − 0.244·67-s + 0.949·71-s + 1.63·73-s − 0.112·79-s + 1.86·83-s − 0.650·85-s − 1.90·89-s + 0.205·95-s − 0.304·97-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

\(L(1)\) \(\approx\) \(1.516542519\)
\(L(\frac12)\) \(\approx\) \(1.516542519\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 17 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−7.940963875965950811187412500021, −7.18647837334672438523571766437, −6.66813384608020900552232584451, −5.68098313333067741075620523599, −5.16194350111874490179185123126, −4.53384136488930654225510718868, −3.48029655211268921609097017103, −2.53384233463331346727195600031, −2.06638425598003929422411599917, −0.59569207132295766078901924349, 0.59569207132295766078901924349, 2.06638425598003929422411599917, 2.53384233463331346727195600031, 3.48029655211268921609097017103, 4.53384136488930654225510718868, 5.16194350111874490179185123126, 5.68098313333067741075620523599, 6.66813384608020900552232584451, 7.18647837334672438523571766437, 7.940963875965950811187412500021

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