Properties

Degree 4
Conductor $ 11^{2} \cdot 233^{2} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·11-s − 16-s − 2·17-s − 2·23-s − 2·31-s + 2·37-s − 2·41-s + 2·43-s + 2·49-s − 2·71-s + 2·73-s − 2·79-s − 81-s + 2·89-s + 2·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯
L(s)  = 1  − 2·11-s − 16-s − 2·17-s − 2·23-s − 2·31-s + 2·37-s − 2·41-s + 2·43-s + 2·49-s − 2·71-s + 2·73-s − 2·79-s − 81-s + 2·89-s + 2·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6568969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6568969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(6568969\)    =    \(11^{2} \cdot 233^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{2563} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 6568969,\ (\ :0, 0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.5020002568$
$L(\frac12)$  $\approx$  $0.5020002568$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{11,\;233\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{11,\;233\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad11$C_1$ \( ( 1 + T )^{2} \)
233$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
3$C_2^2$ \( 1 + T^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2$ \( ( 1 + T + T^{2} )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_2$ \( ( 1 + T + T^{2} )^{2} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_2$ \( ( 1 + T + T^{2} )^{2} \)
37$C_2$ \( ( 1 - T + T^{2} )^{2} \)
41$C_2$ \( ( 1 + T + T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + T^{2} )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_2^2$ \( 1 + T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_2$ \( ( 1 + T + T^{2} )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )^{2} \)
79$C_2$ \( ( 1 + T + T^{2} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )^{2} \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−9.024199886102955549724432401123, −8.966308471757042360398399669638, −8.531493529962721411274021462105, −8.198120055933097180651501151626, −7.56992681696179587969796202837, −7.39169254687569231610171592698, −7.24586468610773280239023326767, −6.50892096593132392100438424575, −6.03133522837937432936503459265, −5.97810237262983055390819926905, −5.28800620688109138976054973666, −5.02538466275586459080282852221, −4.47807619637250736327595134535, −4.02889128610351794862898201543, −3.91096781329385457098304243135, −2.88022517795644054179448284056, −2.64741712167278805447573544247, −2.01167708205642339916872271798, −1.94669311806842822348200736691, −0.42656929302875800099008565060, 0.42656929302875800099008565060, 1.94669311806842822348200736691, 2.01167708205642339916872271798, 2.64741712167278805447573544247, 2.88022517795644054179448284056, 3.91096781329385457098304243135, 4.02889128610351794862898201543, 4.47807619637250736327595134535, 5.02538466275586459080282852221, 5.28800620688109138976054973666, 5.97810237262983055390819926905, 6.03133522837937432936503459265, 6.50892096593132392100438424575, 7.24586468610773280239023326767, 7.39169254687569231610171592698, 7.56992681696179587969796202837, 8.198120055933097180651501151626, 8.531493529962721411274021462105, 8.966308471757042360398399669638, 9.024199886102955549724432401123

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