Properties

Degree 16
Conductor $ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} $
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  − 4·4-s + 10·16-s + 40·43-s − 12·49-s − 20·64-s − 48·79-s + 80·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s − 160·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + 48·196-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 2·4-s + 5/2·16-s + 6.09·43-s − 1.71·49-s − 5/2·64-s − 5.40·79-s + 7.27·121-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 + 3.38·169-s − 12.1·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 24/7·196-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3150} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $5.581011037$
$L(\frac12)$  $\approx$  $5.581011037$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{4} \)
3 \( 1 \)
5 \( 1 \)
7 \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
good11 \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \)
13 \( ( 1 - 22 T^{2} + 259 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 + 29 T^{2} + p^{2} T^{4} )^{4} \)
19 \( ( 1 - 16 T^{2} - 14 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 - 30 T^{2} + 107 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 34 T^{2} + 1411 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{4} \)
41 \( ( 1 + 74 T^{2} + 3931 T^{4} + 74 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 10 T + 109 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \)
47 \( ( 1 + 128 T^{2} + 7714 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 110 T^{2} + 8443 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 46 T^{2} + 5691 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 214 T^{2} + 18691 T^{4} - 214 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 + p T^{2} )^{8} \)
71 \( ( 1 - 20 T^{2} - 2618 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 72 T^{2} + 4754 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 12 T + 144 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \)
83 \( ( 1 + 62 T^{2} + 9739 T^{4} + 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 116 T^{2} + 6406 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 48 T^{2} - 9406 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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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

−3.65372524882039269049222162677, −3.46709892583146723666262580269, −3.19426032668186898644493550524, −3.13903070876658904205211244907, −3.09798031129740401547725106931, −3.04080502365365035679623546003, −3.02367893244242087596773436722, −2.94825695922640576597935204213, −2.85905514017897838061766651256, −2.50416316478804066920834252900, −2.28104005597174552675915997213, −2.18312253267085702600948033631, −2.06679718276531990216365453631, −1.98178421291580285180715174728, −1.94017469584060185662696472561, −1.83889492723172694276081102786, −1.53817887704575025300395803655, −1.29897959054539869270647516556, −1.18189696386556021624067733787, −1.07609726055748953870579048024, −0.850607825659126228785260159621, −0.63480286035640344233160431599, −0.57170301167474019055647450567, −0.46153211081304693235084636242, −0.23740602141061827367530712823, 0.23740602141061827367530712823, 0.46153211081304693235084636242, 0.57170301167474019055647450567, 0.63480286035640344233160431599, 0.850607825659126228785260159621, 1.07609726055748953870579048024, 1.18189696386556021624067733787, 1.29897959054539869270647516556, 1.53817887704575025300395803655, 1.83889492723172694276081102786, 1.94017469584060185662696472561, 1.98178421291580285180715174728, 2.06679718276531990216365453631, 2.18312253267085702600948033631, 2.28104005597174552675915997213, 2.50416316478804066920834252900, 2.85905514017897838061766651256, 2.94825695922640576597935204213, 3.02367893244242087596773436722, 3.04080502365365035679623546003, 3.09798031129740401547725106931, 3.13903070876658904205211244907, 3.19426032668186898644493550524, 3.46709892583146723666262580269, 3.65372524882039269049222162677

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

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