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  + 3·4-s + 2·5-s − 9-s − 12·11-s + 5·16-s + 12·19-s + 6·20-s − 25-s + 4·29-s − 20·31-s − 3·36-s + 4·41-s − 36·44-s − 2·45-s − 49-s − 24·55-s + 16·59-s − 4·61-s + 3·64-s + 20·71-s + 36·76-s − 8·79-s + 10·80-s + 81-s − 12·89-s + 24·95-s + 12·99-s + ⋯
L(s)  = 1  + 3/2·4-s + 0.894·5-s − 1/3·9-s − 3.61·11-s + 5/4·16-s + 2.75·19-s + 1.34·20-s − 1/5·25-s + 0.742·29-s − 3.59·31-s − 1/2·36-s + 0.624·41-s − 5.42·44-s − 0.298·45-s − 1/7·49-s − 3.23·55-s + 2.08·59-s − 0.512·61-s + 3/8·64-s + 2.37·71-s + 4.12·76-s − 0.900·79-s + 1.11·80-s + 1/9·81-s − 1.27·89-s + 2.46·95-s + 1.20·99-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  =  $(4,\ 11025,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.34834$
$L(\frac12)$  $\approx$  $1.34834$
$L(\frac{3}{2})$   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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−14.07765741036902913605419866632, −13.34777610896378407247861108327, −13.03878561228530477495552939210, −12.59576519015341727507158702442, −11.86948020238545760806007327417, −11.27433118061208772548131650751, −10.69192697832386041230196696032, −10.65843719583477371813672800041, −9.704083858777687059690286950373, −9.590614160303390257499736002489, −8.370119229283450604227174471651, −7.77449242194542902577294645997, −7.40285334522283855310331904915, −6.96295836787847011215570227142, −5.66673716415926716727854405042, −5.53092384774540902824306223888, −5.20934549967759721253207672094, −3.31926418578659151487141795523, −2.70173620439600678454391775595, −2.06135419027838012408223731157, 2.06135419027838012408223731157, 2.70173620439600678454391775595, 3.31926418578659151487141795523, 5.20934549967759721253207672094, 5.53092384774540902824306223888, 5.66673716415926716727854405042, 6.96295836787847011215570227142, 7.40285334522283855310331904915, 7.77449242194542902577294645997, 8.370119229283450604227174471651, 9.590614160303390257499736002489, 9.704083858777687059690286950373, 10.65843719583477371813672800041, 10.69192697832386041230196696032, 11.27433118061208772548131650751, 11.86948020238545760806007327417, 12.59576519015341727507158702442, 13.03878561228530477495552939210, 13.34777610896378407247861108327, 14.07765741036902913605419866632

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