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  − 2·5-s − 4·7-s − 2·13-s − 2·17-s − 8·23-s − 25-s + 6·29-s − 8·31-s + 8·35-s − 6·37-s − 2·41-s − 8·47-s + 9·49-s + 6·53-s − 4·59-s + 6·61-s + 4·65-s + 4·67-s + 14·73-s + 4·79-s − 12·83-s + 4·85-s + 6·89-s + 8·91-s + 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s − 0.554·13-s − 0.485·17-s − 1.66·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s + 1.35·35-s − 0.986·37-s − 0.312·41-s − 1.16·47-s + 9/7·49-s + 0.824·53-s − 0.520·59-s + 0.768·61-s + 0.496·65-s + 0.488·67-s + 1.63·73-s + 0.450·79-s − 1.31·83-s + 0.433·85-s + 0.635·89-s + 0.838·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-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 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.40134689804792, −13.80906984162797, −13.44037366194561, −12.67021407214076, −12.48711998684699, −11.97191997828909, −11.48262127447052, −10.89779607618004, −10.18214996116107, −9.915358416019185, −9.425519863932825, −8.704544202389419, −8.319513024724529, −7.631952953651156, −7.175066756035343, −6.608996774310288, −6.199530743051457, −5.515097107141709, −4.853683121652481, −4.134532860447208, −3.635352191496820, −3.267264960046378, −2.411870436097074, −1.833841114842064, −0.5523170968488716, 0, 0.5523170968488716, 1.833841114842064, 2.411870436097074, 3.267264960046378, 3.635352191496820, 4.134532860447208, 4.853683121652481, 5.515097107141709, 6.199530743051457, 6.608996774310288, 7.175066756035343, 7.631952953651156, 8.319513024724529, 8.704544202389419, 9.425519863932825, 9.915358416019185, 10.18214996116107, 10.89779607618004, 11.48262127447052, 11.97191997828909, 12.48711998684699, 12.67021407214076, 13.44037366194561, 13.80906984162797, 14.40134689804792

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