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  − 7·13-s + 7·19-s − 5·25-s + 7·31-s − 37-s − 5·43-s + 14·61-s − 11·67-s − 7·73-s + 13·79-s + 14·97-s + 7·103-s + 17·109-s + ⋯
L(s)  = 1  − 1.94·13-s + 1.60·19-s − 25-s + 1.25·31-s − 0.164·37-s − 0.762·43-s + 1.79·61-s − 1.34·67-s − 0.819·73-s + 1.46·79-s + 1.42·97-s + 0.689·103-s + 1.62·109-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

\(L(1)\) \(\approx\) \(1.599591298\)
\(L(\frac12)\) \(\approx\) \(1.599591298\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−7.73594011226772717557429578921, −7.38837049753910494835648660850, −6.64356021456249165343568157928, −5.74202906155295297295597884206, −5.06671097741570885388825087277, −4.51783710729529372415064306443, −3.46966062525893120155742621808, −2.72502754714260381206626087168, −1.89236012616310691619963980542, −0.62873387229788762149874796515, 0.62873387229788762149874796515, 1.89236012616310691619963980542, 2.72502754714260381206626087168, 3.46966062525893120155742621808, 4.51783710729529372415064306443, 5.06671097741570885388825087277, 5.74202906155295297295597884206, 6.64356021456249165343568157928, 7.38837049753910494835648660850, 7.73594011226772717557429578921

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