Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{2} \cdot 7^{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  − 2·3-s + 4-s − 2·5-s + 2·7-s + 3·9-s + 4·11-s − 2·12-s + 4·15-s − 3·16-s − 4·17-s + 4·19-s − 2·20-s − 4·21-s + 8·23-s + 3·25-s − 4·27-s + 2·28-s − 4·29-s + 12·31-s − 8·33-s − 4·35-s + 3·36-s + 4·37-s − 4·41-s + 4·44-s − 6·45-s + 8·47-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.755·7-s + 9-s + 1.20·11-s − 0.577·12-s + 1.03·15-s − 3/4·16-s − 0.970·17-s + 0.917·19-s − 0.447·20-s − 0.872·21-s + 1.66·23-s + 3/5·25-s − 0.769·27-s + 0.377·28-s − 0.742·29-s + 2.15·31-s − 1.39·33-s − 0.676·35-s + 1/2·36-s + 0.657·37-s − 0.624·41-s + 0.603·44-s − 0.894·45-s + 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(11025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 11025,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.785372$
$L(\frac12)$  $\approx$  $0.785372$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
11$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 20 T + 222 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−13.83105489238817075034959756976, −13.64300893606418748618108877373, −12.78164098571730908744395799935, −12.21462615368365629827364694348, −11.82381989334064317755500600093, −11.25571416032418356121748741329, −11.22830280592174644561216623642, −10.71248176543795167070725589629, −9.782036460362608195092526135262, −9.200136912201529736656132372640, −8.627168896087875885705621690209, −7.82513692382023103813259447991, −7.22411327079707474917124684286, −6.69481828136884877767437788201, −6.28297263652039351876956159723, −5.25936024831167293412664975803, −4.60463390503738808095240317214, −4.14755026085407614211409726178, −2.89131068890819257465193141069, −1.31435688386558938615415944577, 1.31435688386558938615415944577, 2.89131068890819257465193141069, 4.14755026085407614211409726178, 4.60463390503738808095240317214, 5.25936024831167293412664975803, 6.28297263652039351876956159723, 6.69481828136884877767437788201, 7.22411327079707474917124684286, 7.82513692382023103813259447991, 8.627168896087875885705621690209, 9.200136912201529736656132372640, 9.782036460362608195092526135262, 10.71248176543795167070725589629, 11.22830280592174644561216623642, 11.25571416032418356121748741329, 11.82381989334064317755500600093, 12.21462615368365629827364694348, 12.78164098571730908744395799935, 13.64300893606418748618108877373, 13.83105489238817075034959756976

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