Properties

Degree 4
Conductor $ 3^{2} \cdot 7^{2} \cdot 11^{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-s − 2·3-s − 2·4-s + 2·6-s − 2·7-s + 3·8-s + 3·9-s + 4·12-s + 4·13-s + 2·14-s + 16-s − 4·17-s − 3·18-s − 6·19-s + 4·21-s − 4·23-s − 6·24-s − 5·25-s − 4·26-s − 4·27-s + 4·28-s − 10·29-s + 12·31-s − 2·32-s + 4·34-s − 6·36-s + 6·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 4-s + 0.816·6-s − 0.755·7-s + 1.06·8-s + 9-s + 1.15·12-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 0.970·17-s − 0.707·18-s − 1.37·19-s + 0.872·21-s − 0.834·23-s − 1.22·24-s − 25-s − 0.784·26-s − 0.769·27-s + 0.755·28-s − 1.85·29-s + 2.15·31-s − 0.353·32-s + 0.685·34-s − 36-s + 0.986·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6456681 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6456681 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(6456681\)    =    \(3^{2} \cdot 7^{2} \cdot 11^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{2541} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 6456681,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 6 T + 27 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 63 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 10 T + 99 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 2 T + 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$C_4$ \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 154 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 24 T + 290 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 8 T + 174 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−8.724420240516919386783489014798, −8.514288415388442860813879361933, −7.82459885775674357039920418760, −7.81159807287467294909904150150, −7.24693812801127627137230282782, −6.50464694666663581733534558096, −6.46563243408670096258286957513, −6.06938857426823510828172508690, −5.59365314835826669698926045518, −5.43795392521933034685995421633, −4.43604560350728737821489722897, −4.41126713093883700683468618247, −3.98360891822452192730159137124, −3.82755163107651186547127156095, −2.71431353088794845757354849048, −2.41239918951327886656515781669, −1.50731509326314313504178088799, −1.00021499715164530701316714837, 0, 0, 1.00021499715164530701316714837, 1.50731509326314313504178088799, 2.41239918951327886656515781669, 2.71431353088794845757354849048, 3.82755163107651186547127156095, 3.98360891822452192730159137124, 4.41126713093883700683468618247, 4.43604560350728737821489722897, 5.43795392521933034685995421633, 5.59365314835826669698926045518, 6.06938857426823510828172508690, 6.46563243408670096258286957513, 6.50464694666663581733534558096, 7.24693812801127627137230282782, 7.81159807287467294909904150150, 7.82459885775674357039920418760, 8.514288415388442860813879361933, 8.724420240516919386783489014798

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