Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \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  − 2·7-s − 4·13-s − 6·17-s − 4·19-s − 6·23-s − 5·25-s − 6·29-s + 8·31-s + 10·37-s + 6·41-s + 8·43-s + 6·47-s − 3·49-s + 8·61-s + 4·67-s − 6·71-s − 2·73-s − 14·79-s + 12·83-s + 6·89-s + 8·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.755·7-s − 1.10·13-s − 1.45·17-s − 0.917·19-s − 1.25·23-s − 25-s − 1.11·29-s + 1.43·31-s + 1.64·37-s + 0.937·41-s + 1.21·43-s + 0.875·47-s − 3/7·49-s + 1.02·61-s + 0.488·67-s − 0.712·71-s − 0.234·73-s − 1.57·79-s + 1.31·83-s + 0.635·89-s + 0.838·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(69696\)    =    \(2^{6} \cdot 3^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{69696} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 69696,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.50519860595836, −13.85038951891626, −13.25082472628155, −12.98004326389451, −12.48637797956932, −11.81836846986971, −11.51840853709193, −10.80903591600239, −10.34882020059256, −9.731210202420295, −9.424143950861452, −8.918602985402226, −8.142963290823511, −7.759383587107171, −7.159695652149537, −6.541388039168566, −6.064681687161160, −5.691383343067341, −4.724403472761299, −4.206669142054927, −3.945532582959703, −2.873878903310139, −2.354849836450472, −1.974037948713719, −0.6824024615033998, 0, 0.6824024615033998, 1.974037948713719, 2.354849836450472, 2.873878903310139, 3.945532582959703, 4.206669142054927, 4.724403472761299, 5.691383343067341, 6.064681687161160, 6.541388039168566, 7.159695652149537, 7.759383587107171, 8.142963290823511, 8.918602985402226, 9.424143950861452, 9.731210202420295, 10.34882020059256, 10.80903591600239, 11.51840853709193, 11.81836846986971, 12.48637797956932, 12.98004326389451, 13.25082472628155, 13.85038951891626, 14.50519860595836

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