Properties

Degree 4
Conductor $ 2^{4} \cdot 29^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 4·7-s − 9-s + 10·13-s + 12·23-s − 7·25-s − 2·29-s − 8·35-s − 2·45-s − 2·49-s + 10·53-s − 28·59-s + 4·63-s + 20·65-s − 8·67-s − 16·71-s − 8·81-s + 4·83-s − 40·91-s − 16·103-s + 20·107-s + 18·109-s + 24·115-s − 10·117-s + 15·121-s − 26·125-s + 127-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.51·7-s − 1/3·9-s + 2.77·13-s + 2.50·23-s − 7/5·25-s − 0.371·29-s − 1.35·35-s − 0.298·45-s − 2/7·49-s + 1.37·53-s − 3.64·59-s + 0.503·63-s + 2.48·65-s − 0.977·67-s − 1.89·71-s − 8/9·81-s + 0.439·83-s − 4.19·91-s − 1.57·103-s + 1.93·107-s + 1.72·109-s + 2.23·115-s − 0.924·117-s + 1.36·121-s − 2.32·125-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 13456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 13456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(13456\)    =    \(2^{4} \cdot 29^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{13456} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 13456,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.126402568$
$L(\frac12)$  $\approx$  $1.126402568$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;29\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;29\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
29$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 151 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 166 T^{2} + p^{2} T^{4} \)
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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

−11.34773848136740402885803148359, −10.56442536522991048073397422435, −10.25475190170627145434171778233, −9.399564312813529369362579074415, −9.090378683942619914044714318202, −8.742874493889395478290563940354, −7.87529861398798680728963547354, −7.06487408227525129793152854033, −6.35570806238188015824100744160, −6.02071832667706757065760761607, −5.61983243082601145682414998739, −4.43739594559277891528076962187, −3.39739732224506275679364929632, −3.08808685831935640145183700853, −1.50679528450003839441860314788, 1.50679528450003839441860314788, 3.08808685831935640145183700853, 3.39739732224506275679364929632, 4.43739594559277891528076962187, 5.61983243082601145682414998739, 6.02071832667706757065760761607, 6.35570806238188015824100744160, 7.06487408227525129793152854033, 7.87529861398798680728963547354, 8.742874493889395478290563940354, 9.090378683942619914044714318202, 9.399564312813529369362579074415, 10.25475190170627145434171778233, 10.56442536522991048073397422435, 11.34773848136740402885803148359

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