Properties

Degree 4
Conductor $ 3^{6} \cdot 5^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·4-s − 2·5-s − 2·7-s − 3·8-s − 2·10-s − 4·11-s + 5·13-s − 2·14-s + 16-s − 17-s − 3·19-s + 4·20-s − 4·22-s − 13·23-s + 3·25-s + 5·26-s + 4·28-s − 29-s + 2·32-s − 34-s + 4·35-s − 3·38-s + 6·40-s − 5·41-s − 2·43-s + 8·44-s + ⋯
L(s)  = 1  + 0.707·2-s − 4-s − 0.894·5-s − 0.755·7-s − 1.06·8-s − 0.632·10-s − 1.20·11-s + 1.38·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.688·19-s + 0.894·20-s − 0.852·22-s − 2.71·23-s + 3/5·25-s + 0.980·26-s + 0.755·28-s − 0.185·29-s + 0.353·32-s − 0.171·34-s + 0.676·35-s − 0.486·38-s + 0.948·40-s − 0.780·41-s − 0.304·43-s + 1.20·44-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(893025\)    =    \(3^{6} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{945} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 893025,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 5 T + 31 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + T + 33 T^{2} + p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 13 T + 87 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + T - 43 T^{2} + p T^{3} + p^{2} T^{4} \)
31$V_4$ \( 1 + 17 T^{2} + p^{2} T^{4} \)
37$V_4$ \( 1 + 29 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 5 T + 57 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 11 T + 135 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 16 T + 137 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 21 T + 221 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 7 T + 85 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + T + 131 T^{2} + p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 10 T + 91 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 9 T + 177 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 137 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 6 T + 167 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 9 T + 153 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
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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

−9.834814892749257393832066142655, −9.406703153240767885191554916019, −8.991227256322384876700815324382, −8.455068163802194315339698757933, −8.177582226516844944927988643487, −7.920222104281531654170329335234, −7.44388713694371732256281233477, −6.62652671638674137004661111524, −6.19647478656595924400803890664, −6.10549484652247078006660884337, −5.29129357206834833475015060053, −4.96678938605535477894573788617, −4.38999148689490299031447079082, −4.03079214007870853271298107808, −3.47150343482780152789955580828, −3.33569908990296030142128982539, −2.40997672140956335910460485092, −1.58618687098928773813721097511, 0, 0, 1.58618687098928773813721097511, 2.40997672140956335910460485092, 3.33569908990296030142128982539, 3.47150343482780152789955580828, 4.03079214007870853271298107808, 4.38999148689490299031447079082, 4.96678938605535477894573788617, 5.29129357206834833475015060053, 6.10549484652247078006660884337, 6.19647478656595924400803890664, 6.62652671638674137004661111524, 7.44388713694371732256281233477, 7.920222104281531654170329335234, 8.177582226516844944927988643487, 8.455068163802194315339698757933, 8.991227256322384876700815324382, 9.406703153240767885191554916019, 9.834814892749257393832066142655

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