Properties

Degree 8
Conductor $ 5^{4} \cdot 11^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·9-s − 8·16-s + 18·23-s + 25-s + 4·27-s − 14·37-s − 24·47-s + 16·48-s − 12·53-s + 26·67-s − 36·69-s − 12·71-s − 2·75-s − 3·81-s − 34·97-s + 8·103-s + 28·111-s − 42·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 48·141-s − 16·144-s + 149-s + ⋯
L(s)  = 1  − 1.15·3-s + 2/3·9-s − 2·16-s + 3.75·23-s + 1/5·25-s + 0.769·27-s − 2.30·37-s − 3.50·47-s + 2.30·48-s − 1.64·53-s + 3.17·67-s − 4.33·69-s − 1.42·71-s − 0.230·75-s − 1/3·81-s − 3.45·97-s + 0.788·103-s + 2.65·111-s − 3.95·113-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.04·141-s − 4/3·144-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(9150625\)    =    \(5^{4} \cdot 11^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{55} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 9150625,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.386846$
$L(\frac12)$  $\approx$  $0.386846$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;11\}$, \(F_p(T)\) is a polynomial of degree 8. If $p \in \{5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \)
3$C_2$$\times$$C_2^2$ \( ( 1 + T + p T^{2} )^{2}( 1 - 5 T^{2} + p^{2} T^{4} ) \)
7$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2$$\times$$C_2^2$ \( ( 1 - 9 T + p T^{2} )^{2}( 1 + 35 T^{2} + p^{2} T^{4} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$$\times$$C_2^2$ \( ( 1 + 7 T + p T^{2} )^{2}( 1 - 25 T^{2} + p^{2} T^{4} ) \)
41$C_2$ \( ( 1 - p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
47$C_2$$\times$$C_2^2$ \( ( 1 + 12 T + p T^{2} )^{2}( 1 + 50 T^{2} + p^{2} T^{4} ) \)
53$C_2$$\times$$C_2^2$ \( ( 1 + 6 T + p T^{2} )^{2}( 1 - 70 T^{2} + p^{2} T^{4} ) \)
59$C_2$ \( ( 1 - 15 T + p T^{2} )^{2}( 1 + 15 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - p T^{2} )^{4} \)
67$C_2$$\times$$C_2^2$ \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 35 T^{2} + p^{2} T^{4} ) \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$$\times$$C_2^2$ \( ( 1 + 17 T + p T^{2} )^{2}( 1 + 95 T^{2} + p^{2} T^{4} ) \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.48835924044194409813873677265, −11.06498465641040350573303438256, −10.88138989192355342832196302463, −10.80504971422507305312653999049, −10.53863727231625196018524193354, −9.704271736043385903489154722322, −9.682303072155755329859385347862, −9.444252028395126395372971013627, −8.813174501455526830475471380197, −8.796465424989386929398989851816, −8.278663877808189586967777173745, −8.083045099302244928948101749412, −7.17128803580651590665461554075, −7.15190906861610606750797165916, −6.70998683208428034530949216352, −6.60354026877879547142645090208, −6.31419279479992790599764988959, −5.38481502662120442712443357971, −5.18591295389126206683905173421, −4.90586862298137220451443600131, −4.69283492377536908539427929709, −3.94953060036178666445540372141, −3.14810190884510550788584453401, −2.85991361577896401400185087078, −1.63620250690544105536133033406, 1.63620250690544105536133033406, 2.85991361577896401400185087078, 3.14810190884510550788584453401, 3.94953060036178666445540372141, 4.69283492377536908539427929709, 4.90586862298137220451443600131, 5.18591295389126206683905173421, 5.38481502662120442712443357971, 6.31419279479992790599764988959, 6.60354026877879547142645090208, 6.70998683208428034530949216352, 7.15190906861610606750797165916, 7.17128803580651590665461554075, 8.083045099302244928948101749412, 8.278663877808189586967777173745, 8.796465424989386929398989851816, 8.813174501455526830475471380197, 9.444252028395126395372971013627, 9.682303072155755329859385347862, 9.704271736043385903489154722322, 10.53863727231625196018524193354, 10.80504971422507305312653999049, 10.88138989192355342832196302463, 11.06498465641040350573303438256, 11.48835924044194409813873677265

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