Properties

Degree 10
Conductor $ 2^{15} \cdot 7^{5} \cdot 11^{5} \cdot 13^{5} $
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  + 3-s − 5-s + 5·7-s − 4·9-s − 5·11-s − 5·13-s − 15-s + 19·17-s − 5·19-s + 5·21-s + 5·23-s − 6·25-s − 9·27-s − 17·29-s + 4·31-s − 5·33-s − 5·35-s + 10·37-s − 5·39-s + 10·41-s − 25·43-s + 4·45-s + 3·47-s + 15·49-s + 19·51-s + 22·53-s + 5·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1.88·7-s − 4/3·9-s − 1.50·11-s − 1.38·13-s − 0.258·15-s + 4.60·17-s − 1.14·19-s + 1.09·21-s + 1.04·23-s − 6/5·25-s − 1.73·27-s − 3.15·29-s + 0.718·31-s − 0.870·33-s − 0.845·35-s + 1.64·37-s − 0.800·39-s + 1.56·41-s − 3.81·43-s + 0.596·45-s + 0.437·47-s + 15/7·49-s + 2.66·51-s + 3.02·53-s + 0.674·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{15} \cdot 7^{5} \cdot 11^{5} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{15} \cdot 7^{5} \cdot 11^{5} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(10\)
\( N \)  =  \(2^{15} \cdot 7^{5} \cdot 11^{5} \cdot 13^{5}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8008} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(10,\ 2^{15} \cdot 7^{5} \cdot 11^{5} \cdot 13^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $8.438188948$
$L(\frac12)$  $\approx$  $8.438188948$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11,\;13\}$, \(F_p(T)\) is a polynomial of degree 10. If $p \in \{2,\;7,\;11,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 9.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
7$C_1$ \( ( 1 - T )^{5} \)
11$C_1$ \( ( 1 + T )^{5} \)
13$C_1$ \( ( 1 + T )^{5} \)
good3$C_2 \wr S_5$ \( 1 - T + 5 T^{2} + 11 T^{4} + 14 T^{5} + 11 p T^{6} + 5 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + T + 7 T^{2} + 4 T^{3} + 19 T^{4} + 19 p T^{6} + 4 p^{2} T^{7} + 7 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 19 T + 211 T^{2} - 1636 T^{3} + 9643 T^{4} - 44660 T^{5} + 9643 p T^{6} - 1636 p^{2} T^{7} + 211 p^{3} T^{8} - 19 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 5 T + 65 T^{2} + 260 T^{3} + 2113 T^{4} + 6722 T^{5} + 2113 p T^{6} + 260 p^{2} T^{7} + 65 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 5 T + 79 T^{2} - 466 T^{3} + 2923 T^{4} - 16082 T^{5} + 2923 p T^{6} - 466 p^{2} T^{7} + 79 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 17 T + 173 T^{2} + 1010 T^{3} + 4519 T^{4} + 18428 T^{5} + 4519 p T^{6} + 1010 p^{2} T^{7} + 173 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 4 T + 77 T^{2} - 512 T^{3} + 2801 T^{4} - 24388 T^{5} + 2801 p T^{6} - 512 p^{2} T^{7} + 77 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 10 T + 125 T^{2} - 585 T^{3} + 4387 T^{4} - 14264 T^{5} + 4387 p T^{6} - 585 p^{2} T^{7} + 125 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 10 T + 93 T^{2} - 15 p T^{3} + 4027 T^{4} - 29860 T^{5} + 4027 p T^{6} - 15 p^{3} T^{7} + 93 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 25 T + 8 p T^{2} + 3373 T^{3} + 27232 T^{4} + 190396 T^{5} + 27232 p T^{6} + 3373 p^{2} T^{7} + 8 p^{4} T^{8} + 25 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 3 T + 137 T^{2} - 818 T^{3} + 8733 T^{4} - 62242 T^{5} + 8733 p T^{6} - 818 p^{2} T^{7} + 137 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 22 T + 412 T^{2} - 4808 T^{3} + 49834 T^{4} - 382030 T^{5} + 49834 p T^{6} - 4808 p^{2} T^{7} + 412 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 21 T + 186 T^{2} - 1247 T^{3} + 15310 T^{4} - 157544 T^{5} + 15310 p T^{6} - 1247 p^{2} T^{7} + 186 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 26 T + 547 T^{2} - 121 p T^{3} + 83563 T^{4} - 708520 T^{5} + 83563 p T^{6} - 121 p^{3} T^{7} + 547 p^{3} T^{8} - 26 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 28 T + 545 T^{2} - 7368 T^{3} + 82657 T^{4} - 735404 T^{5} + 82657 p T^{6} - 7368 p^{2} T^{7} + 545 p^{3} T^{8} - 28 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 28 T + 563 T^{2} - 7840 T^{3} + 1286 p T^{4} - 835592 T^{5} + 1286 p^{2} T^{6} - 7840 p^{2} T^{7} + 563 p^{3} T^{8} - 28 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 4 T + 193 T^{2} + 29 T^{3} + 15799 T^{4} - 36904 T^{5} + 15799 p T^{6} + 29 p^{2} T^{7} + 193 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 11 T + 182 T^{2} + 1077 T^{3} + 22036 T^{4} + 150844 T^{5} + 22036 p T^{6} + 1077 p^{2} T^{7} + 182 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 193 T^{2} - 195 T^{3} + 20681 T^{4} - 45478 T^{5} + 20681 p T^{6} - 195 p^{2} T^{7} + 193 p^{3} T^{8} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 37 T + 945 T^{2} + 16086 T^{3} + 218857 T^{4} + 2276992 T^{5} + 218857 p T^{6} + 16086 p^{2} T^{7} + 945 p^{3} T^{8} + 37 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 18 T + 469 T^{2} - 6621 T^{3} + 89573 T^{4} - 946052 T^{5} + 89573 p T^{6} - 6621 p^{2} T^{7} + 469 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.60538966098885951867697288679, −4.40901474017149812705305060317, −4.08344876876554845633130017447, −4.00815060493492054520001230669, −3.93721307920865937044992452339, −3.89687989572956842627452935644, −3.79011844500480376859152741415, −3.46971637601482508249585324428, −3.46053202266129969278695975108, −3.19836531745547061472953756429, −2.78047353624407192800325537743, −2.76450242933116034936924820803, −2.71397492234039464088391002515, −2.55417936202686031582718895452, −2.50508175826909749611089912421, −2.05997796194135093907712092708, −1.99599700706837008135856648721, −1.85523199076702321991460043564, −1.67088108786584385522571126638, −1.50276820321548057916392226687, −0.997931210659447750062113828744, −0.890433851686605851861420779703, −0.75236697924852741213140753057, −0.48455994677648237936127141533, −0.28007954631653532598378553397, 0.28007954631653532598378553397, 0.48455994677648237936127141533, 0.75236697924852741213140753057, 0.890433851686605851861420779703, 0.997931210659447750062113828744, 1.50276820321548057916392226687, 1.67088108786584385522571126638, 1.85523199076702321991460043564, 1.99599700706837008135856648721, 2.05997796194135093907712092708, 2.50508175826909749611089912421, 2.55417936202686031582718895452, 2.71397492234039464088391002515, 2.76450242933116034936924820803, 2.78047353624407192800325537743, 3.19836531745547061472953756429, 3.46053202266129969278695975108, 3.46971637601482508249585324428, 3.79011844500480376859152741415, 3.89687989572956842627452935644, 3.93721307920865937044992452339, 4.00815060493492054520001230669, 4.08344876876554845633130017447, 4.40901474017149812705305060317, 4.60538966098885951867697288679

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