# 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 of factors

## 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.