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 − 6·11-s − 3·13-s − 4·17-s + 5·19-s − 4·23-s − 25-s + 4·29-s − 7·31-s − 9·37-s + 2·41-s + 43-s + 2·47-s − 8·53-s + 12·55-s + 10·61-s + 6·65-s + 15·67-s − 6·71-s − 11·73-s − 79-s + 6·83-s + 8·85-s + 8·89-s − 10·95-s − 14·97-s + 6·101-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.80·11-s − 0.832·13-s − 0.970·17-s + 1.14·19-s − 0.834·23-s − 1/5·25-s + 0.742·29-s − 1.25·31-s − 1.47·37-s + 0.312·41-s + 0.152·43-s + 0.291·47-s − 1.09·53-s + 1.61·55-s + 1.28·61-s + 0.744·65-s + 1.83·67-s − 0.712·71-s − 1.28·73-s − 0.112·79-s + 0.658·83-s + 0.867·85-s + 0.847·89-s − 1.02·95-s − 1.42·97-s + 0.597·101-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: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5454159124\)
\(L(\frac12)\) \(\approx\) \(0.5454159124\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−7.86837664713197532679773648533, −7.39014768213158472645987411444, −6.75849057276581334970677589340, −5.60638508100849073652067189206, −5.15172024971649584397733102179, −4.39380155779821844339831789631, −3.54736352653234231503567437326, −2.74026869503774083685113957342, −1.96283649374467943530461796317, −0.35120958609157001347548789487, 0.35120958609157001347548789487, 1.96283649374467943530461796317, 2.74026869503774083685113957342, 3.54736352653234231503567437326, 4.39380155779821844339831789631, 5.15172024971649584397733102179, 5.60638508100849073652067189206, 6.75849057276581334970677589340, 7.39014768213158472645987411444, 7.86837664713197532679773648533

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