Properties

Degree 4
Conductor $ 2^{12} \cdot 3^{2} \cdot 7^{4} $
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  − 2·3-s + 3·9-s + 8·19-s − 10·25-s − 4·27-s + 12·29-s − 16·31-s − 4·37-s − 16·47-s + 4·53-s − 16·57-s + 8·59-s + 20·75-s + 5·81-s − 24·83-s − 24·87-s + 32·93-s − 16·103-s + 12·109-s + 8·111-s + 4·113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 32·141-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s + 1.83·19-s − 2·25-s − 0.769·27-s + 2.22·29-s − 2.87·31-s − 0.657·37-s − 2.33·47-s + 0.549·53-s − 2.11·57-s + 1.04·59-s + 2.30·75-s + 5/9·81-s − 2.63·83-s − 2.57·87-s + 3.31·93-s − 1.57·103-s + 1.14·109-s + 0.759·111-s + 0.376·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.69·141-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(88510464\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{9408} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((4,\ 88510464,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + p T^{2} )^{2} \)
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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

−7.38763552863504066914788983011, −7.21728830975177467499149622850, −6.74729704993067497746749882969, −6.66180757883093863062879148413, −5.97861324327630514907772698902, −5.83199696764183515886776601216, −5.41251345812635484926475327871, −5.34404050852531286507562869394, −4.70578945366403615049158522883, −4.64992658910754387744825163086, −3.86397682604065271570758608836, −3.83287944982740122886710933174, −3.20394493316144183179453579989, −3.04184520970393234090360317844, −2.17069110891621622709579115728, −1.97285604311821539407399239925, −1.23677812021589812783982408537, −1.10927131514791689287240561276, 0, 0, 1.10927131514791689287240561276, 1.23677812021589812783982408537, 1.97285604311821539407399239925, 2.17069110891621622709579115728, 3.04184520970393234090360317844, 3.20394493316144183179453579989, 3.83287944982740122886710933174, 3.86397682604065271570758608836, 4.64992658910754387744825163086, 4.70578945366403615049158522883, 5.34404050852531286507562869394, 5.41251345812635484926475327871, 5.83199696764183515886776601216, 5.97861324327630514907772698902, 6.66180757883093863062879148413, 6.74729704993067497746749882969, 7.21728830975177467499149622850, 7.38763552863504066914788983011

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