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  − 2·13-s + 8·19-s − 5·25-s − 4·31-s − 10·37-s − 8·43-s − 14·61-s + 16·67-s + 10·73-s + 4·79-s − 14·97-s + 20·103-s + 2·109-s + ⋯
L(s)  = 1  − 0.554·13-s + 1.83·19-s − 25-s − 0.718·31-s − 1.64·37-s − 1.21·43-s − 1.79·61-s + 1.95·67-s + 1.17·73-s + 0.450·79-s − 1.42·97-s + 1.97·103-s + 0.191·109-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: \(1\)
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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
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

−7.55348034334582611202573667047, −7.01678164134878036064700533521, −6.19926344367899340966030638418, −5.29098134252625465872380406245, −4.99119743714055460794390382794, −3.79768693875931379182391471043, −3.27428940642722031982966070935, −2.24166501763825187887414431892, −1.32729943198718475773059677845, 0, 1.32729943198718475773059677845, 2.24166501763825187887414431892, 3.27428940642722031982966070935, 3.79768693875931379182391471043, 4.99119743714055460794390382794, 5.29098134252625465872380406245, 6.19926344367899340966030638418, 7.01678164134878036064700533521, 7.55348034334582611202573667047

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