Properties

Degree 8
Conductor $ 2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4} $
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  − 8·13-s − 16-s + 12·17-s + 8·23-s − 24·29-s − 24·31-s + 4·37-s + 20·43-s + 20·47-s − 8·53-s − 32·59-s + 32·61-s − 4·67-s − 24·73-s − 16·83-s − 8·89-s − 16·97-s + 8·103-s + 8·107-s + 16·113-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 2.21·13-s − 1/4·16-s + 2.91·17-s + 1.66·23-s − 4.45·29-s − 4.31·31-s + 0.657·37-s + 3.04·43-s + 2.91·47-s − 1.09·53-s − 4.16·59-s + 4.09·61-s − 0.488·67-s − 2.80·73-s − 1.75·83-s − 0.847·89-s − 1.62·97-s + 0.788·103-s + 0.773·107-s + 1.50·113-s + 1.81·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3150} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $2.647729106$
$L(\frac12)$  $\approx$  $2.647729106$
$L(\frac{3}{2})$   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 8. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2^2$ \( 1 + T^{4} \)
3 \( 1 \)
5 \( 1 \)
7$C_2^2$ \( 1 + T^{4} \)
good11$D_4\times C_2$ \( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 + 8 T + 32 T^{2} + 136 T^{3} + 562 T^{4} + 136 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 12 T + 72 T^{2} - 372 T^{3} + 1726 T^{4} - 372 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 8 T + 32 T^{2} - 120 T^{3} + 386 T^{4} - 120 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 4 T + 8 T^{2} + 244 T^{3} - 2162 T^{4} + 244 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 - 140 T^{2} + 8134 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 20 T + 200 T^{2} - 1780 T^{3} + 13726 T^{4} - 1780 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 20 T + 200 T^{2} - 1860 T^{3} + 15182 T^{4} - 1860 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 8 T + 32 T^{2} + 360 T^{3} + 3986 T^{4} + 360 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
61$D_{4}$ \( ( 1 - 16 T + 184 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 4 T + 8 T^{2} + 260 T^{3} + 8446 T^{4} + 260 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 120 T^{2} + 9074 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 52 T^{2} + 11110 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 + 16 T + 128 T^{2} + 816 T^{3} + 4178 T^{4} + 816 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 16 T + 128 T^{2} + 1488 T^{3} + 17282 T^{4} + 1488 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.86788169274077302047597449761, −5.84604812713721242148296533872, −5.69457033357114548571134118197, −5.56815889093120248179875097669, −5.37446865885741896620333997673, −5.31623076142919796900468160979, −5.18075396463620687253856508172, −4.48001283313385336046514333623, −4.44772554609412277626013988481, −4.43640181956095379047313711904, −4.21574381892640133108339444670, −3.65284204727890436855739749492, −3.56550488057576299344553345313, −3.48593459473262998273581175468, −3.37351232724570179105632955338, −2.74990076614155504863510322201, −2.73638249043570679269708135855, −2.61622580916779869568255226057, −2.15088657271021988618501190948, −1.75725168677132275018266508608, −1.73929204026715908590831042123, −1.53175132943956846750161837661, −1.00004516500147651667628390211, −0.46611510454590898647007515716, −0.37819456128522002662235569882, 0.37819456128522002662235569882, 0.46611510454590898647007515716, 1.00004516500147651667628390211, 1.53175132943956846750161837661, 1.73929204026715908590831042123, 1.75725168677132275018266508608, 2.15088657271021988618501190948, 2.61622580916779869568255226057, 2.73638249043570679269708135855, 2.74990076614155504863510322201, 3.37351232724570179105632955338, 3.48593459473262998273581175468, 3.56550488057576299344553345313, 3.65284204727890436855739749492, 4.21574381892640133108339444670, 4.43640181956095379047313711904, 4.44772554609412277626013988481, 4.48001283313385336046514333623, 5.18075396463620687253856508172, 5.31623076142919796900468160979, 5.37446865885741896620333997673, 5.56815889093120248179875097669, 5.69457033357114548571134118197, 5.84604812713721242148296533872, 5.86788169274077302047597449761

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