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  − 5.29·11-s + 5.29·23-s − 5·25-s + 10.5·29-s + 6·37-s − 12·43-s + 10.5·53-s − 4·67-s − 5.29·71-s − 8·79-s − 5.29·107-s − 18·109-s − 21.1·113-s + ⋯
L(s)  = 1  − 1.59·11-s + 1.10·23-s − 25-s + 1.96·29-s + 0.986·37-s − 1.82·43-s + 1.45·53-s − 0.488·67-s − 0.627·71-s − 0.900·79-s − 0.511·107-s − 1.72·109-s − 1.99·113-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 \( 1 + 5T^{2} \)
11 \( 1 + 5.29T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 5.29T + 23T^{2} \)
29 \( 1 - 10.5T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 5.29T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−7.68046967421742912538433333907, −6.92368915862081844712231084358, −6.20791150142134488291686373787, −5.34127051865836217130018379303, −4.88953720900795964389067858473, −4.00703789716805152785371293092, −2.94123026085743864659582152393, −2.49393734490330670576404426053, −1.23892001217160548574782403808, 0, 1.23892001217160548574782403808, 2.49393734490330670576404426053, 2.94123026085743864659582152393, 4.00703789716805152785371293092, 4.88953720900795964389067858473, 5.34127051865836217130018379303, 6.20791150142134488291686373787, 6.92368915862081844712231084358, 7.68046967421742912538433333907

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