Properties

Degree 4
Conductor $ 3^{2} \cdot 7^{3} \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 3·4-s − 4·5-s − 7-s + 9-s − 6·12-s − 8·15-s + 5·16-s + 8·17-s + 12·20-s − 2·21-s + 2·25-s − 4·27-s + 3·28-s + 4·35-s − 3·36-s − 12·37-s + 8·41-s + 24·43-s − 4·45-s − 20·47-s + 10·48-s + 49-s + 16·51-s + 4·59-s + 24·60-s − 63-s + ⋯
L(s)  = 1  + 1.15·3-s − 3/2·4-s − 1.78·5-s − 0.377·7-s + 1/3·9-s − 1.73·12-s − 2.06·15-s + 5/4·16-s + 1.94·17-s + 2.68·20-s − 0.436·21-s + 2/5·25-s − 0.769·27-s + 0.566·28-s + 0.676·35-s − 1/2·36-s − 1.97·37-s + 1.24·41-s + 3.65·43-s − 0.596·45-s − 2.91·47-s + 1.44·48-s + 1/7·49-s + 2.24·51-s + 0.520·59-s + 3.09·60-s − 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373527 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373527 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(373527\)    =    \(3^{2} \cdot 7^{3} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{373527} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((4,\ 373527,\ (\ :1/2, 1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_1$ \( 1 + T \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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\[\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.503382678912186991908631693316, −7.959279630374951452453810048684, −7.82867506007281568614416512666, −7.42272354779304253907064109454, −6.84576801400653059993297260469, −5.96443978110022033983848919969, −5.47259073254467597109793181667, −5.02042250101516742575983822668, −4.11624652535699675148140948571, −4.07930883034575373640745113861, −3.42834085160890094357341585879, −3.26888344602981049454952325554, −2.30344485967966511448007334997, −1.01039812063869528093138870562, 0, 1.01039812063869528093138870562, 2.30344485967966511448007334997, 3.26888344602981049454952325554, 3.42834085160890094357341585879, 4.07930883034575373640745113861, 4.11624652535699675148140948571, 5.02042250101516742575983822668, 5.47259073254467597109793181667, 5.96443978110022033983848919969, 6.84576801400653059993297260469, 7.42272354779304253907064109454, 7.82867506007281568614416512666, 7.959279630374951452453810048684, 8.503382678912186991908631693316

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