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  + 1.41·5-s − 4·11-s + 4.24·13-s − 7.07·17-s + 5.65·19-s − 8·23-s − 2.99·25-s + 2·29-s + 4·37-s + 9.89·41-s + 4·43-s − 5.65·47-s + 4·53-s − 5.65·55-s + 11.3·59-s + 1.41·61-s + 6·65-s + 12·67-s − 15.5·73-s + 16·79-s + 5.65·83-s − 10.0·85-s + 7.07·89-s + 8.00·95-s + 7.07·97-s + 12.7·101-s − 5.65·103-s + ⋯
L(s)  = 1  + 0.632·5-s − 1.20·11-s + 1.17·13-s − 1.71·17-s + 1.29·19-s − 1.66·23-s − 0.599·25-s + 0.371·29-s + 0.657·37-s + 1.54·41-s + 0.609·43-s − 0.825·47-s + 0.549·53-s − 0.762·55-s + 1.47·59-s + 0.181·61-s + 0.744·65-s + 1.46·67-s − 1.82·73-s + 1.80·79-s + 0.620·83-s − 1.08·85-s + 0.749·89-s + 0.820·95-s + 0.717·97-s + 1.26·101-s − 0.557·103-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  =  \((2,\ 7056,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.981750321\)
\(L(\frac12)\)  \(\approx\)  \(1.981750321\)
\(L(\frac{3}{2})\)   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 \( 1 - 1.41T + 5T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + 7.07T + 17T^{2} \)
19 \( 1 - 5.65T + 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 - 9.89T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 - 1.41T + 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 15.5T + 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 - 5.65T + 83T^{2} \)
89 \( 1 - 7.07T + 89T^{2} \)
97 \( 1 - 7.07T + 97T^{2} \)
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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

−7.969211615481351891843541775523, −7.32255808568705907177971119939, −6.32301369786361087113965678331, −5.93918724800471733548830556486, −5.21644729068976999674154883021, −4.34696937816104427224743853008, −3.60654459229780335061141967597, −2.51951359637012108384803085445, −1.99231613645070061606911589619, −0.70026480379897881243970751324, 0.70026480379897881243970751324, 1.99231613645070061606911589619, 2.51951359637012108384803085445, 3.60654459229780335061141967597, 4.34696937816104427224743853008, 5.21644729068976999674154883021, 5.93918724800471733548830556486, 6.32301369786361087113965678331, 7.32255808568705907177971119939, 7.969211615481351891843541775523

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