Properties

Degree 4
Conductor $ 2^{12} \cdot 3^{2} \cdot 7^{4} $
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 + 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  =  \(0\)
Selberg data  =  \((4,\ 88510464,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(5.833796394\)
\(L(\frac12)\)  \(\approx\)  \(5.833796394\)
\(L(\frac{3}{2})\)   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

−7.911933362660355096301638688843, −7.67753911174006134563695587785, −7.30051105559858652801059203182, −6.82672441438608402066648718749, −6.40322208328912833402246525687, −6.33209716339531137071855047537, −6.02751448165583504117786541015, −5.48216527177464350845805590794, −4.94112180930035082106382725068, −4.63340131782117377259318855332, −4.32562635880199084549945882466, −4.01754468724643796816698474815, −3.72093086885127854385664539478, −3.10873618733757644875261117089, −2.75346007989027902441743287998, −2.51021804242624827538827399585, −1.95245090885116493869228210470, −1.78724827875133163800567208756, −0.857457788768882232748049337839, −0.60888953459971999994697655639, 0.60888953459971999994697655639, 0.857457788768882232748049337839, 1.78724827875133163800567208756, 1.95245090885116493869228210470, 2.51021804242624827538827399585, 2.75346007989027902441743287998, 3.10873618733757644875261117089, 3.72093086885127854385664539478, 4.01754468724643796816698474815, 4.32562635880199084549945882466, 4.63340131782117377259318855332, 4.94112180930035082106382725068, 5.48216527177464350845805590794, 6.02751448165583504117786541015, 6.33209716339531137071855047537, 6.40322208328912833402246525687, 6.82672441438608402066648718749, 7.30051105559858652801059203182, 7.67753911174006134563695587785, 7.911933362660355096301638688843

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