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  − 2.82·5-s + 6·11-s − 5.65·13-s − 1.41·17-s + 4.24·19-s + 4·23-s + 3.00·25-s + 6·29-s − 2.82·31-s + 2·37-s + 1.41·41-s − 10·43-s − 2.82·47-s + 2·53-s − 16.9·55-s + 1.41·59-s − 8.48·61-s + 16.0·65-s − 4·67-s − 12·71-s − 9.89·73-s + 4·79-s + 1.41·83-s + 4.00·85-s + 4.24·89-s − 12·95-s + 12.7·97-s + ⋯
L(s)  = 1  − 1.26·5-s + 1.80·11-s − 1.56·13-s − 0.342·17-s + 0.973·19-s + 0.834·23-s + 0.600·25-s + 1.11·29-s − 0.508·31-s + 0.328·37-s + 0.220·41-s − 1.52·43-s − 0.412·47-s + 0.274·53-s − 2.28·55-s + 0.184·59-s − 1.08·61-s + 1.98·65-s − 0.488·67-s − 1.42·71-s − 1.15·73-s + 0.450·79-s + 0.155·83-s + 0.433·85-s + 0.449·89-s − 1.23·95-s + 1.29·97-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  =  \((2,\ 7056,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.395011138\)
\(L(\frac12)\)  \(\approx\)  \(1.395011138\)
\(L(\frac{3}{2})\)   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 \( 1 + 2.82T + 5T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 - 4.24T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 2.82T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 1.41T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 1.41T + 59T^{2} \)
61 \( 1 + 8.48T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 9.89T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 1.41T + 83T^{2} \)
89 \( 1 - 4.24T + 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−7.82632224636268707738844512862, −7.17671109456718182259662692951, −6.83599943364003155639979165473, −5.89996236354136227367083182381, −4.75681735409154423647814216043, −4.50341937202362248346770819380, −3.52902903902973866602280651938, −2.96865577631073130815384270789, −1.69263622372052197459451285730, −0.61227819689763455025954233183, 0.61227819689763455025954233183, 1.69263622372052197459451285730, 2.96865577631073130815384270789, 3.52902903902973866602280651938, 4.50341937202362248346770819380, 4.75681735409154423647814216043, 5.89996236354136227367083182381, 6.83599943364003155639979165473, 7.17671109456718182259662692951, 7.82632224636268707738844512862

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