Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 7 \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s + 7-s + 9-s − 6·13-s − 2·15-s − 2·17-s + 4·19-s + 21-s − 25-s + 27-s + 2·29-s − 8·31-s − 2·35-s + 6·37-s − 6·39-s − 10·41-s − 4·43-s − 2·45-s + 8·47-s + 49-s − 2·51-s + 6·53-s + 4·57-s − 4·59-s + 10·61-s + 63-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.66·13-s − 0.516·15-s − 0.485·17-s + 0.917·19-s + 0.218·21-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.338·35-s + 0.986·37-s − 0.960·39-s − 1.56·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s + 0.529·57-s − 0.520·59-s + 1.28·61-s + 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(40656\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{40656} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 40656,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.97604097102439, −14.67217958681097, −14.01191538875626, −13.57889854865678, −12.91945027799115, −12.37286126958121, −11.85293252241708, −11.56008466980058, −10.86258553333696, −10.23081221583596, −9.703905894679903, −9.197804657807305, −8.630013699579708, −7.886945374438881, −7.692335698581459, −7.079694817561442, −6.635834844392715, −5.560172240438497, −5.056374565047149, −4.539757157321770, −3.784202602546271, −3.371644950630349, −2.423070679705750, −2.054356506242279, −0.9166070569173201, 0, 0.9166070569173201, 2.054356506242279, 2.423070679705750, 3.371644950630349, 3.784202602546271, 4.539757157321770, 5.056374565047149, 5.560172240438497, 6.635834844392715, 7.079694817561442, 7.692335698581459, 7.886945374438881, 8.630013699579708, 9.197804657807305, 9.703905894679903, 10.23081221583596, 10.86258553333696, 11.56008466980058, 11.85293252241708, 12.37286126958121, 12.91945027799115, 13.57889854865678, 14.01191538875626, 14.67217958681097, 14.97604097102439

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