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  − 0.585·5-s − 0.828·11-s − 1.41·13-s − 2.24·17-s − 6.82·19-s + 4.82·23-s − 4.65·25-s − 8.48·29-s − 5.17·31-s + 1.65·37-s + 0.585·41-s + 8·43-s + 6.82·47-s + 13.3·53-s + 0.485·55-s + 5.17·59-s + 13.8·61-s + 0.828·65-s + 8·67-s − 0.828·71-s − 11.0·73-s + 2.34·79-s + 15.3·83-s + 1.31·85-s + 10.7·89-s + 4·95-s − 7.75·97-s + ⋯
L(s)  = 1  − 0.261·5-s − 0.249·11-s − 0.392·13-s − 0.543·17-s − 1.56·19-s + 1.00·23-s − 0.931·25-s − 1.57·29-s − 0.928·31-s + 0.272·37-s + 0.0914·41-s + 1.21·43-s + 0.996·47-s + 1.82·53-s + 0.0654·55-s + 0.673·59-s + 1.77·61-s + 0.102·65-s + 0.977·67-s − 0.0983·71-s − 1.29·73-s + 0.263·79-s + 1.68·83-s + 0.142·85-s + 1.13·89-s + 0.410·95-s − 0.787·97-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

\(L(1)\) \(\approx\) \(1.265442711\)
\(L(\frac12)\) \(\approx\) \(1.265442711\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.585T + 5T^{2} \)
11 \( 1 + 0.828T + 11T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 + 2.24T + 17T^{2} \)
19 \( 1 + 6.82T + 19T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 + 8.48T + 29T^{2} \)
31 \( 1 + 5.17T + 31T^{2} \)
37 \( 1 - 1.65T + 37T^{2} \)
41 \( 1 - 0.585T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 6.82T + 47T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 - 5.17T + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 0.828T + 71T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 - 2.34T + 79T^{2} \)
83 \( 1 - 15.3T + 83T^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 + 7.75T + 97T^{2} \)
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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

−7.84559382173864157893131293476, −7.27535810676448797876926330767, −6.65237302822365979181137794291, −5.74235152952742374041253616764, −5.21169154175944496043081488782, −4.13978158786338060007048669129, −3.82652219361121838120568632629, −2.53774985939799479818192528144, −2.00266703295827614311349636913, −0.54688236611422705431268569970, 0.54688236611422705431268569970, 2.00266703295827614311349636913, 2.53774985939799479818192528144, 3.82652219361121838120568632629, 4.13978158786338060007048669129, 5.21169154175944496043081488782, 5.74235152952742374041253616764, 6.65237302822365979181137794291, 7.27535810676448797876926330767, 7.84559382173864157893131293476

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