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  − 1.41·5-s − 4·11-s − 4.24·13-s + 7.07·17-s − 5.65·19-s − 8·23-s − 2.99·25-s + 2·29-s + 4·37-s − 9.89·41-s + 4·43-s + 5.65·47-s + 4·53-s + 5.65·55-s − 11.3·59-s − 1.41·61-s + 6·65-s + 12·67-s + 15.5·73-s + 16·79-s − 5.65·83-s − 10.0·85-s − 7.07·89-s + 8.00·95-s − 7.07·97-s − 12.7·101-s + 5.65·103-s + ⋯
L(s)  = 1  − 0.632·5-s − 1.20·11-s − 1.17·13-s + 1.71·17-s − 1.29·19-s − 1.66·23-s − 0.599·25-s + 0.371·29-s + 0.657·37-s − 1.54·41-s + 0.609·43-s + 0.825·47-s + 0.549·53-s + 0.762·55-s − 1.47·59-s − 0.181·61-s + 0.744·65-s + 1.46·67-s + 1.82·73-s + 1.80·79-s − 0.620·83-s − 1.08·85-s − 0.749·89-s + 0.820·95-s − 0.717·97-s − 1.26·101-s + 0.557·103-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: $\chi_{7056} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8568698294\)
\(L(\frac12)\) \(\approx\) \(0.8568698294\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.41T + 5T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 - 7.07T + 17T^{2} \)
19 \( 1 + 5.65T + 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 9.89T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 1.41T + 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 15.5T + 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 + 5.65T + 83T^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 + 7.07T + 97T^{2} \)
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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

−8.000986055808044704131864045510, −7.47886674305724041792151187021, −6.57810833253111255509064167535, −5.73132766061274245836021446321, −5.15180032184453151614871203081, −4.31886659857370845762107413181, −3.62569051648689026834325333295, −2.67698996935937688268733183432, −1.96332279105174111741697194843, −0.44492244615190841629398251278, 0.44492244615190841629398251278, 1.96332279105174111741697194843, 2.67698996935937688268733183432, 3.62569051648689026834325333295, 4.31886659857370845762107413181, 5.15180032184453151614871203081, 5.73132766061274245836021446321, 6.57810833253111255509064167535, 7.47886674305724041792151187021, 8.000986055808044704131864045510

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