Properties

Degree 4
Conductor $ 2^{6} \cdot 3^{2} \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·9-s − 2·11-s − 8·13-s − 6·23-s − 25-s − 5·27-s − 2·33-s − 2·37-s − 8·39-s − 10·49-s + 6·59-s − 8·61-s − 6·69-s + 30·71-s − 8·73-s − 75-s + 81-s + 12·83-s − 14·97-s + 4·99-s + 12·107-s + 4·109-s − 2·111-s + 16·117-s + 3·121-s + 127-s + ⋯
L(s)  = 1  + 0.577·3-s − 2/3·9-s − 0.603·11-s − 2.21·13-s − 1.25·23-s − 1/5·25-s − 0.962·27-s − 0.348·33-s − 0.328·37-s − 1.28·39-s − 1.42·49-s + 0.781·59-s − 1.02·61-s − 0.722·69-s + 3.56·71-s − 0.936·73-s − 0.115·75-s + 1/9·81-s + 1.31·83-s − 1.42·97-s + 0.402·99-s + 1.16·107-s + 0.383·109-s − 0.189·111-s + 1.47·117-s + 3/11·121-s + 0.0887·127-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(69696\)    =    \(2^{6} \cdot 3^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{69696} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((4,\ 69696,\ (\ :1/2, 1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( 1 - T + p T^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
71$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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\[\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.726921937470959380626935593622, −9.205754524823445845175580209741, −8.408943998062523444280533863335, −8.132665807481574488057539990847, −7.60852684015633966980596452192, −7.22282278046792198667036847650, −6.47645013269452670867443390425, −5.90441514720049446274076604978, −5.11227132195889762323685903220, −4.92614337906569626690399921734, −3.98508995562062136618490430533, −3.27381985666804460223319703873, −2.49109929152055119200232183305, −2.07797879851041904182307638354, 0, 2.07797879851041904182307638354, 2.49109929152055119200232183305, 3.27381985666804460223319703873, 3.98508995562062136618490430533, 4.92614337906569626690399921734, 5.11227132195889762323685903220, 5.90441514720049446274076604978, 6.47645013269452670867443390425, 7.22282278046792198667036847650, 7.60852684015633966980596452192, 8.132665807481574488057539990847, 8.408943998062523444280533863335, 9.205754524823445845175580209741, 9.726921937470959380626935593622

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