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 − 11-s − 2·13-s + 3·17-s + 5·19-s − 3·23-s − 4·25-s + 6·29-s − 31-s − 5·37-s − 10·41-s + 4·43-s − 47-s + 9·53-s + 55-s − 3·59-s − 3·61-s + 2·65-s − 11·67-s + 16·71-s − 7·73-s + 11·79-s + 4·83-s − 3·85-s − 9·89-s − 5·95-s − 6·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s − 0.554·13-s + 0.727·17-s + 1.14·19-s − 0.625·23-s − 4/5·25-s + 1.11·29-s − 0.179·31-s − 0.821·37-s − 1.56·41-s + 0.609·43-s − 0.145·47-s + 1.23·53-s + 0.134·55-s − 0.390·59-s − 0.384·61-s + 0.248·65-s − 1.34·67-s + 1.89·71-s − 0.819·73-s + 1.23·79-s + 0.439·83-s − 0.325·85-s − 0.953·89-s − 0.512·95-s − 0.609·97-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: $\chi_{7056} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
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 \( 1 + T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−17.56257679404264, −16.74037516181441, −16.34941440713862, −15.60446656665638, −15.29711261195277, −14.50609450483386, −13.79566019797932, −13.58937537372646, −12.45868512921091, −12.11192710860115, −11.66889171314934, −10.83650870487352, −10.07783882286296, −9.782309744840260, −8.866174029853945, −8.166000491811737, −7.602337349494099, −7.061385115580659, −6.170217175746112, −5.385561630117208, −4.839360904592445, −3.867315123366753, −3.234596149378530, −2.331279839773358, −1.233552966095351, 0, 1.233552966095351, 2.331279839773358, 3.234596149378530, 3.867315123366753, 4.839360904592445, 5.385561630117208, 6.170217175746112, 7.061385115580659, 7.602337349494099, 8.166000491811737, 8.866174029853945, 9.782309744840260, 10.07783882286296, 10.83650870487352, 11.66889171314934, 12.11192710860115, 12.45868512921091, 13.58937537372646, 13.79566019797932, 14.50609450483386, 15.29711261195277, 15.60446656665638, 16.34941440713862, 16.74037516181441, 17.56257679404264

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