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  − 3.41·5-s − 2·11-s + 2.58·13-s + 2.24·17-s + 2.82·19-s − 7.65·23-s + 6.65·25-s + 6.82·29-s + 1.17·31-s − 4·37-s − 6.24·41-s − 5.65·43-s − 2.82·47-s + 2·53-s + 6.82·55-s − 1.17·59-s + 12.2·61-s − 8.82·65-s + 5.65·67-s + 9.31·71-s + 13.8·73-s − 13.6·79-s + 7.31·83-s − 7.65·85-s + 14.2·89-s − 9.65·95-s + 2.58·97-s + ⋯
L(s)  = 1  − 1.52·5-s − 0.603·11-s + 0.717·13-s + 0.543·17-s + 0.648·19-s − 1.59·23-s + 1.33·25-s + 1.26·29-s + 0.210·31-s − 0.657·37-s − 0.974·41-s − 0.862·43-s − 0.412·47-s + 0.274·53-s + 0.920·55-s − 0.152·59-s + 1.56·61-s − 1.09·65-s + 0.691·67-s + 1.10·71-s + 1.62·73-s − 1.53·79-s + 0.802·83-s − 0.830·85-s + 1.50·89-s − 0.990·95-s + 0.262·97-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: \(1\)
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 \( 1 + 3.41T + 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 2.58T + 13T^{2} \)
17 \( 1 - 2.24T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 + 7.65T + 23T^{2} \)
29 \( 1 - 6.82T + 29T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 1.17T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 - 9.31T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 - 7.31T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 - 2.58T + 97T^{2} \)
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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

−7.78448862739273330815218123613, −6.96829823455528476656538533109, −6.31669332329476244176907301385, −5.33075773874890257026782941947, −4.73314714532105852460388231046, −3.70208936368141272254606387191, −3.50482297221063585092269416635, −2.37022443881284216851369091798, −1.08560958480826197450945713323, 0, 1.08560958480826197450945713323, 2.37022443881284216851369091798, 3.50482297221063585092269416635, 3.70208936368141272254606387191, 4.73314714532105852460388231046, 5.33075773874890257026782941947, 6.31669332329476244176907301385, 6.96829823455528476656538533109, 7.78448862739273330815218123613

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