Properties

Degree 4
Conductor $ 3^{2} \cdot 7^{4} \cdot 41^{2} $
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 − 2·4-s − 4·5-s + 3·9-s + 2·11-s + 4·12-s − 4·13-s + 8·15-s − 2·17-s + 8·19-s + 8·20-s + 4·25-s − 4·27-s + 2·29-s + 6·31-s − 4·33-s − 6·36-s − 2·37-s + 8·39-s + 2·41-s − 10·43-s − 4·44-s − 12·45-s − 18·47-s + 4·51-s + 8·52-s + 8·53-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 1.78·5-s + 9-s + 0.603·11-s + 1.15·12-s − 1.10·13-s + 2.06·15-s − 0.485·17-s + 1.83·19-s + 1.78·20-s + 4/5·25-s − 0.769·27-s + 0.371·29-s + 1.07·31-s − 0.696·33-s − 36-s − 0.328·37-s + 1.28·39-s + 0.312·41-s − 1.52·43-s − 0.603·44-s − 1.78·45-s − 2.62·47-s + 0.560·51-s + 1.10·52-s + 1.09·53-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 36324729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 36324729 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(36324729\)    =    \(3^{2} \cdot 7^{4} \cdot 41^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6027} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 36324729,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;7,\;41\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
41$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 18 T + 173 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 2 T + 91 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 6 T + 101 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 12 T + 152 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 24 T + 320 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
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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

−7.81536128051432299388540779343, −7.68542676738369525551702232502, −7.09605216694609755127372990738, −6.84727915170172598621879760082, −6.62297992810949274203209241665, −6.23042952138944208166638268726, −5.50118210440411029203220875502, −5.32623041096835230558497133466, −4.90542458372635834329249302918, −4.70513986997764905516749695502, −4.29492636100223364127288075987, −4.03332972064782946531072227623, −3.41170808101923195852329295083, −3.38508879765128773443871102874, −2.65769814446075140339979762284, −2.00559952499735569126155279467, −1.26491025786676716303416602889, −0.819393996952786040339090110820, 0, 0, 0.819393996952786040339090110820, 1.26491025786676716303416602889, 2.00559952499735569126155279467, 2.65769814446075140339979762284, 3.38508879765128773443871102874, 3.41170808101923195852329295083, 4.03332972064782946531072227623, 4.29492636100223364127288075987, 4.70513986997764905516749695502, 4.90542458372635834329249302918, 5.32623041096835230558497133466, 5.50118210440411029203220875502, 6.23042952138944208166638268726, 6.62297992810949274203209241665, 6.84727915170172598621879760082, 7.09605216694609755127372990738, 7.68542676738369525551702232502, 7.81536128051432299388540779343

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