Properties

Degree 4
Conductor $ 13 \cdot 29^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 5-s − 6·7-s − 4·9-s + 5·13-s − 3·16-s + 20-s − 3·23-s − 3·25-s + 6·28-s − 29-s + 6·35-s + 4·36-s + 4·45-s + 13·49-s − 5·52-s − 8·53-s − 8·59-s + 24·63-s + 7·64-s − 5·65-s + 11·67-s + 20·71-s + 3·80-s + 7·81-s + 18·83-s − 30·91-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.447·5-s − 2.26·7-s − 4/3·9-s + 1.38·13-s − 3/4·16-s + 0.223·20-s − 0.625·23-s − 3/5·25-s + 1.13·28-s − 0.185·29-s + 1.01·35-s + 2/3·36-s + 0.596·45-s + 13/7·49-s − 0.693·52-s − 1.09·53-s − 1.04·59-s + 3.02·63-s + 7/8·64-s − 0.620·65-s + 1.34·67-s + 2.37·71-s + 0.335·80-s + 7/9·81-s + 1.97·83-s − 3.14·91-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 10933 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 10933 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(10933\)    =    \(13 \cdot 29^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{10933} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 10933,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{13,\;29\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{13,\;29\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 4 T + p T^{2} ) \)
29$C_2$ \( 1 + T + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 - 5 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
89$C_2^2$ \( 1 + 132 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.00274500823544086006338237784, −10.82208890856886567537829541902, −9.802565269717364274116980350787, −9.432442717053843467843421263055, −9.080752251714741804081934892647, −8.290154120354716418454238759078, −7.951550605171018050800626255731, −6.77365830292056409944570659155, −6.39148144486217093494604677284, −5.93918958836021597292803536270, −5.08977600180795895939868749989, −3.82546822223363121729442740600, −3.57055238928682927694670032397, −2.63525639291551106854450960432, 0, 2.63525639291551106854450960432, 3.57055238928682927694670032397, 3.82546822223363121729442740600, 5.08977600180795895939868749989, 5.93918958836021597292803536270, 6.39148144486217093494604677284, 6.77365830292056409944570659155, 7.951550605171018050800626255731, 8.290154120354716418454238759078, 9.080752251714741804081934892647, 9.432442717053843467843421263055, 9.802565269717364274116980350787, 10.82208890856886567537829541902, 11.00274500823544086006338237784

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