Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.74·5-s + 5.29·11-s + 4.24·13-s − 3.74·17-s + 2.82·19-s − 5.29·23-s + 9·25-s + 5.29·29-s + 8.48·31-s + 4·37-s − 3.74·41-s − 8·43-s − 7.48·47-s + 10.5·53-s − 19.7·55-s − 7.48·59-s − 9.89·61-s − 15.8·65-s − 12·67-s + 15.8·71-s − 1.41·73-s + 4·79-s − 14.9·83-s + 14·85-s + 3.74·89-s − 10.5·95-s + 9.89·97-s + ⋯
L(s)  = 1  − 1.67·5-s + 1.59·11-s + 1.17·13-s − 0.907·17-s + 0.648·19-s − 1.10·23-s + 1.80·25-s + 0.982·29-s + 1.52·31-s + 0.657·37-s − 0.584·41-s − 1.21·43-s − 1.09·47-s + 1.45·53-s − 2.66·55-s − 0.974·59-s − 1.26·61-s − 1.96·65-s − 1.46·67-s + 1.88·71-s − 0.165·73-s + 0.450·79-s − 1.64·83-s + 1.51·85-s + 0.396·89-s − 1.08·95-s + 1.00·97-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

\( d \)  =  \(2\)
\( N \)  =  \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7056} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 7056,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.534544039\)
\(L(\frac12)\)  \(\approx\)  \(1.534544039\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 3.74T + 5T^{2} \)
11 \( 1 - 5.29T + 11T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + 3.74T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 + 5.29T + 23T^{2} \)
29 \( 1 - 5.29T + 29T^{2} \)
31 \( 1 - 8.48T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 3.74T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 7.48T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + 7.48T + 59T^{2} \)
61 \( 1 + 9.89T + 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 - 15.8T + 71T^{2} \)
73 \( 1 + 1.41T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 14.9T + 83T^{2} \)
89 \( 1 - 3.74T + 89T^{2} \)
97 \( 1 - 9.89T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.108274622340933570374065894845, −7.24398593246117225230484190713, −6.50585525894888119760597319127, −6.16392751090425258792929268176, −4.77055487390027247736608970514, −4.27422892936224152891349808595, −3.66640374138027896867439052608, −3.03057701853497052232525851509, −1.58999074059699369503867861727, −0.66777016501471800554745949790, 0.66777016501471800554745949790, 1.58999074059699369503867861727, 3.03057701853497052232525851509, 3.66640374138027896867439052608, 4.27422892936224152891349808595, 4.77055487390027247736608970514, 6.16392751090425258792929268176, 6.50585525894888119760597319127, 7.24398593246117225230484190713, 8.108274622340933570374065894845

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