Properties

Degree 4
Conductor $ 2^{6} \cdot 11^{4} $
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  + 2·2-s − 2·3-s + 2·4-s − 4·6-s − 3·9-s − 4·12-s − 4·16-s + 4·17-s − 6·18-s − 9·25-s + 14·27-s − 8·32-s + 8·34-s − 6·36-s + 16·41-s + 12·43-s + 8·48-s − 10·49-s − 18·50-s − 8·51-s + 28·54-s + 10·59-s − 8·64-s − 14·67-s + 8·68-s − 8·73-s + 18·75-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 4-s − 1.63·6-s − 9-s − 1.15·12-s − 16-s + 0.970·17-s − 1.41·18-s − 9/5·25-s + 2.69·27-s − 1.41·32-s + 1.37·34-s − 36-s + 2.49·41-s + 1.82·43-s + 1.15·48-s − 1.42·49-s − 2.54·50-s − 1.12·51-s + 3.81·54-s + 1.30·59-s − 64-s − 1.71·67-s + 0.970·68-s − 0.936·73-s + 2.07·75-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 937024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 937024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(937024\)    =    \(2^{6} \cdot 11^{4}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{937024} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 937024,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 - p T + p T^{2} \)
11 \( 1 \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.78789607043645765017922373874, −7.45239985706761183706670864930, −6.76841176385697439798678244963, −6.23870064789214498217427813253, −5.91635357600802194053301753892, −5.72624020676726467506368629260, −5.41175071356617719099292784896, −4.73179154047144907827706428783, −4.44574490854809981150981932393, −3.73048318654850784959880571386, −3.35845173002445544137333678583, −2.63044898935838963010520851422, −2.28590628922737670281570587392, −1.01639287860097037078534087292, 0, 1.01639287860097037078534087292, 2.28590628922737670281570587392, 2.63044898935838963010520851422, 3.35845173002445544137333678583, 3.73048318654850784959880571386, 4.44574490854809981150981932393, 4.73179154047144907827706428783, 5.41175071356617719099292784896, 5.72624020676726467506368629260, 5.91635357600802194053301753892, 6.23870064789214498217427813253, 6.76841176385697439798678244963, 7.45239985706761183706670864930, 7.78789607043645765017922373874

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