Properties

Degree 16
Conductor $ 3^{8} \cdot 5^{16} \cdot 7^{8} $
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  − 24·11-s + 8·16-s − 36·31-s + 60·61-s − 48·71-s + 81-s + 260·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 192·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 7.23·11-s + 2·16-s − 6.46·31-s + 7.68·61-s − 5.69·71-s + 1/9·81-s + 23.6·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s − 14.4·176-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(3^{8} \cdot 5^{16} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{525} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 3^{8} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $0.489426$
$L(\frac12)$  $\approx$  $0.489426$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad3 \( 1 - T^{4} + T^{8} \)
5 \( 1 \)
7 \( 1 - 94 T^{4} + p^{4} T^{8} \)
good2 \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )^{2}( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2} \)
11 \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \)
13 \( ( 1 + 146 T^{4} + p^{4} T^{8} )^{2} \)
17 \( ( 1 - 8 T + 32 T^{2} + 16 T^{3} - 353 T^{4} + 16 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )( 1 + 8 T + 32 T^{2} - 16 T^{3} - 353 T^{4} - 16 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} ) \)
19 \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2}( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \)
23 \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - p T^{2} )^{8} \)
31 \( ( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{4} \)
37 \( 1 + 2737 T^{4} + 5617008 T^{8} + 2737 p^{4} T^{12} + p^{8} T^{16} \)
41 \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \)
43 \( ( 1 - 217 T^{4} + p^{4} T^{8} )^{2} \)
47 \( 1 + 1054 T^{4} - 3768765 T^{8} + 1054 p^{4} T^{12} + p^{8} T^{16} \)
53 \( 1 + 5614 T^{4} + 23626515 T^{8} + 5614 p^{4} T^{12} + p^{8} T^{16} \)
59 \( ( 1 - 10 T^{2} - 3381 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 14 T + p T^{2} )^{4}( 1 - T + p T^{2} )^{4} \)
67 \( ( 1 - 8809 T^{4} + p^{4} T^{8} )( 1 + 2903 T^{4} + p^{4} T^{8} ) \)
71 \( ( 1 + 6 T + p T^{2} )^{8} \)
73 \( 1 - 10367 T^{4} + 79076448 T^{8} - 10367 p^{4} T^{12} + p^{8} T^{16} \)
79 \( ( 1 + 133 T^{2} + 11448 T^{4} + 133 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 3122 T^{4} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 18193 T^{4} + p^{4} T^{8} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.99391307161124015396356665960, −4.58174004981262061270803164047, −4.55516690837679482890276387157, −4.52423455768890817243656155478, −4.19489773233159367548112308722, −3.89747800404006669862979237514, −3.84656270687051038701794293506, −3.82753760399692837185009775914, −3.51564662680074924093609276015, −3.38845253616131406567043228967, −3.25391789292548959715191719530, −3.22816677713087200743535681577, −2.91367530668217284890669431162, −2.69957126160765208841230825974, −2.65253997425464728676013197441, −2.52048140508014136373237582391, −2.41663640039743755558624071599, −2.07940811170143476097044867019, −1.97719339403107622471735206980, −1.90850787529175323334230912388, −1.47925147411753385254175920294, −1.46398499375201067928077764017, −0.61901554323764199639669016041, −0.52425848865602382620456819157, −0.21278174363786797361557722989, 0.21278174363786797361557722989, 0.52425848865602382620456819157, 0.61901554323764199639669016041, 1.46398499375201067928077764017, 1.47925147411753385254175920294, 1.90850787529175323334230912388, 1.97719339403107622471735206980, 2.07940811170143476097044867019, 2.41663640039743755558624071599, 2.52048140508014136373237582391, 2.65253997425464728676013197441, 2.69957126160765208841230825974, 2.91367530668217284890669431162, 3.22816677713087200743535681577, 3.25391789292548959715191719530, 3.38845253616131406567043228967, 3.51564662680074924093609276015, 3.82753760399692837185009775914, 3.84656270687051038701794293506, 3.89747800404006669862979237514, 4.19489773233159367548112308722, 4.52423455768890817243656155478, 4.55516690837679482890276387157, 4.58174004981262061270803164047, 4.99391307161124015396356665960

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

Plot not available for L-functions of degree greater than 10.