Properties

Degree 4
Conductor $ 2^{4} \cdot 7^{2} \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 2·7-s − 2·11-s + 4·13-s + 4·17-s + 8·23-s + 2·25-s − 4·29-s + 8·31-s − 8·35-s + 8·37-s − 4·41-s − 4·43-s − 8·47-s + 3·49-s − 8·55-s − 4·61-s + 16·65-s − 16·71-s + 20·73-s + 4·77-s − 12·79-s − 9·81-s − 24·83-s + 16·85-s + 12·89-s − 8·91-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.755·7-s − 0.603·11-s + 1.10·13-s + 0.970·17-s + 1.66·23-s + 2/5·25-s − 0.742·29-s + 1.43·31-s − 1.35·35-s + 1.31·37-s − 0.624·41-s − 0.609·43-s − 1.16·47-s + 3/7·49-s − 1.07·55-s − 0.512·61-s + 1.98·65-s − 1.89·71-s + 2.34·73-s + 0.455·77-s − 1.35·79-s − 81-s − 2.63·83-s + 1.73·85-s + 1.27·89-s − 0.838·91-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 94864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 94864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(94864\)    =    \(2^{4} \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{308} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 94864,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.04709$
$L(\frac12)$  $\approx$  $2.04709$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + 104 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 72 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 16 T + 182 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 20 T + 240 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 24 T + 286 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 20 T + 270 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
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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

−11.79862544556652117258581879534, −11.48394573060304426569075730966, −10.93268980626317002630002221542, −10.32595941868223752647292328970, −9.954959392489633817784112235944, −9.807840157894385176075972635581, −9.177677313425725443745857663632, −8.803488770478684976386538125621, −8.159319437625624284725585915287, −7.65766783860508522896836152125, −6.92091491351518040446086260932, −6.46071100623735997890262068396, −5.84969430511133985288320968834, −5.75832604746648542077413438749, −5.04231410927294119410876844444, −4.34243817203963430244828267007, −3.28513107251763483330736154463, −2.99018439790270799030868777916, −2.04106593022041204613406247911, −1.19119964338012444166540234020, 1.19119964338012444166540234020, 2.04106593022041204613406247911, 2.99018439790270799030868777916, 3.28513107251763483330736154463, 4.34243817203963430244828267007, 5.04231410927294119410876844444, 5.75832604746648542077413438749, 5.84969430511133985288320968834, 6.46071100623735997890262068396, 6.92091491351518040446086260932, 7.65766783860508522896836152125, 8.159319437625624284725585915287, 8.803488770478684976386538125621, 9.177677313425725443745857663632, 9.807840157894385176075972635581, 9.954959392489633817784112235944, 10.32595941868223752647292328970, 10.93268980626317002630002221542, 11.48394573060304426569075730966, 11.79862544556652117258581879534

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