Properties

Degree 10
Conductor $ 2^{10} \cdot 3^{5} \cdot 167^{5} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 5

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·3-s − 7·5-s − 2·7-s + 15·9-s − 5·11-s − 8·13-s − 35·15-s − 7·17-s + 2·19-s − 10·21-s − 13·23-s + 13·25-s + 35·27-s − 11·29-s − 12·31-s − 25·33-s + 14·35-s − 7·37-s − 40·39-s − 12·41-s − 105·45-s − 19·47-s − 20·49-s − 35·51-s − 21·53-s + 35·55-s + 10·57-s + ⋯
L(s)  = 1  + 2.88·3-s − 3.13·5-s − 0.755·7-s + 5·9-s − 1.50·11-s − 2.21·13-s − 9.03·15-s − 1.69·17-s + 0.458·19-s − 2.18·21-s − 2.71·23-s + 13/5·25-s + 6.73·27-s − 2.04·29-s − 2.15·31-s − 4.35·33-s + 2.36·35-s − 1.15·37-s − 6.40·39-s − 1.87·41-s − 15.6·45-s − 2.77·47-s − 2.85·49-s − 4.90·51-s − 2.88·53-s + 4.71·55-s + 1.32·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{5} \cdot 167^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{5} \cdot 167^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(10\)
\( N \)  =  \(2^{10} \cdot 3^{5} \cdot 167^{5}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{2004} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(5\)
Selberg data  =  \((10,\ 2^{10} \cdot 3^{5} \cdot 167^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;167\}$,\(F_p(T)\) is a polynomial of degree 10. If $p \in \{2,\;3,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 9.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 - T )^{5} \)
167$C_1$ \( ( 1 + T )^{5} \)
good5$C_2 \wr S_5$ \( 1 + 7 T + 36 T^{2} + 134 T^{3} + 394 T^{4} + 981 T^{5} + 394 p T^{6} + 134 p^{2} T^{7} + 36 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 2 T + 24 T^{2} + 41 T^{3} + 286 T^{4} + 405 T^{5} + 286 p T^{6} + 41 p^{2} T^{7} + 24 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 5 T + 36 T^{2} + 136 T^{3} + 628 T^{4} + 2025 T^{5} + 628 p T^{6} + 136 p^{2} T^{7} + 36 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 8 T + 64 T^{2} + 292 T^{3} + 1433 T^{4} + 4915 T^{5} + 1433 p T^{6} + 292 p^{2} T^{7} + 64 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 7 T + 79 T^{2} + 402 T^{3} + 2557 T^{4} + 9625 T^{5} + 2557 p T^{6} + 402 p^{2} T^{7} + 79 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 2 T + 50 T^{2} - 86 T^{3} + 1389 T^{4} - 2425 T^{5} + 1389 p T^{6} - 86 p^{2} T^{7} + 50 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 13 T + 144 T^{2} + 1052 T^{3} + 6778 T^{4} + 34395 T^{5} + 6778 p T^{6} + 1052 p^{2} T^{7} + 144 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 11 T + 166 T^{2} + 1174 T^{3} + 9958 T^{4} + 49617 T^{5} + 9958 p T^{6} + 1174 p^{2} T^{7} + 166 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 12 T + 153 T^{2} + 1291 T^{3} + 9382 T^{4} + 57399 T^{5} + 9382 p T^{6} + 1291 p^{2} T^{7} + 153 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 7 T + 178 T^{2} + 920 T^{3} + 12752 T^{4} + 48883 T^{5} + 12752 p T^{6} + 920 p^{2} T^{7} + 178 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 12 T + 193 T^{2} + 1363 T^{3} + 13060 T^{4} + 69271 T^{5} + 13060 p T^{6} + 1363 p^{2} T^{7} + 193 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 107 T^{2} - 197 T^{3} + 6754 T^{4} - 10885 T^{5} + 6754 p T^{6} - 197 p^{2} T^{7} + 107 p^{3} T^{8} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 19 T + 168 T^{2} + 1313 T^{3} + 12955 T^{4} + 108573 T^{5} + 12955 p T^{6} + 1313 p^{2} T^{7} + 168 p^{3} T^{8} + 19 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 21 T + 350 T^{2} + 4071 T^{3} + 40465 T^{4} + 314151 T^{5} + 40465 p T^{6} + 4071 p^{2} T^{7} + 350 p^{3} T^{8} + 21 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 7 T + 196 T^{2} + 789 T^{3} + 15775 T^{4} + 43795 T^{5} + 15775 p T^{6} + 789 p^{2} T^{7} + 196 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 6 T + 158 T^{2} + 1262 T^{3} + 14037 T^{4} + 105179 T^{5} + 14037 p T^{6} + 1262 p^{2} T^{7} + 158 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 10 T + 142 T^{2} - 236 T^{3} - 205 T^{4} + 59767 T^{5} - 205 p T^{6} - 236 p^{2} T^{7} + 142 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 35 T + 763 T^{2} + 11235 T^{3} + 130765 T^{4} + 1205501 T^{5} + 130765 p T^{6} + 11235 p^{2} T^{7} + 763 p^{3} T^{8} + 35 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 8 T + 312 T^{2} + 2092 T^{3} + 42635 T^{4} + 220341 T^{5} + 42635 p T^{6} + 2092 p^{2} T^{7} + 312 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 197 T^{2} + 805 T^{3} + 17746 T^{4} + 119621 T^{5} + 17746 p T^{6} + 805 p^{2} T^{7} + 197 p^{3} T^{8} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 11 T + 326 T^{2} + 2557 T^{3} + 45301 T^{4} + 272481 T^{5} + 45301 p T^{6} + 2557 p^{2} T^{7} + 326 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 32 T + 606 T^{2} + 7762 T^{3} + 881 p T^{4} + 741105 T^{5} + 881 p^{2} T^{6} + 7762 p^{2} T^{7} + 606 p^{3} T^{8} + 32 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 11 T + 505 T^{2} - 4153 T^{3} + 99605 T^{4} - 598793 T^{5} + 99605 p T^{6} - 4153 p^{2} T^{7} + 505 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.74018722167512119730204308119, −5.63763563342599919484299263966, −5.46909155274823512957530644414, −5.46428084943718152382330818906, −5.17272213531317762740741372467, −4.68048858821666511717350231746, −4.66980537293146552541935556462, −4.63182857947169563066807306593, −4.39497368902159792431797994453, −4.34838078365187692253764561309, −4.05279710399523654098746584766, −3.71656765480256856522208934026, −3.67715217455151757843915691562, −3.66311810024425602446962900452, −3.49467658768158127292921290469, −3.07600684168264929405032420356, −3.03358858914212224293941068619, −2.91646278961716228305228875553, −2.65136481485911717227223386597, −2.64197031437183232825877177375, −1.95022783321261367404790550879, −1.83992559203755456349688191873, −1.73789437158128062451700941297, −1.71105587019226403451880842145, −1.59539316550349475434123811209, 0, 0, 0, 0, 0, 1.59539316550349475434123811209, 1.71105587019226403451880842145, 1.73789437158128062451700941297, 1.83992559203755456349688191873, 1.95022783321261367404790550879, 2.64197031437183232825877177375, 2.65136481485911717227223386597, 2.91646278961716228305228875553, 3.03358858914212224293941068619, 3.07600684168264929405032420356, 3.49467658768158127292921290469, 3.66311810024425602446962900452, 3.67715217455151757843915691562, 3.71656765480256856522208934026, 4.05279710399523654098746584766, 4.34838078365187692253764561309, 4.39497368902159792431797994453, 4.63182857947169563066807306593, 4.66980537293146552541935556462, 4.68048858821666511717350231746, 5.17272213531317762740741372467, 5.46428084943718152382330818906, 5.46909155274823512957530644414, 5.63763563342599919484299263966, 5.74018722167512119730204308119

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