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Properties

Label 10-2583e5-1.1-c1e5-0-0
Degree $10$
Conductor $1.150\times 10^{17}$
Sign $1$
Analytic cond. $3.73256\times 10^{6}$
Root an. cond. $4.54151$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·4-s + 5·5-s + 5·7-s − 3·8-s + 5·10-s − 2·11-s + 5·13-s + 5·14-s + 3·16-s − 13·17-s − 15·20-s − 2·22-s − 2·23-s + 11·25-s + 5·26-s − 15·28-s + 5·29-s + 17·31-s + 4·32-s − 13·34-s + 25·35-s − 7·37-s − 15·40-s + 5·41-s + 43-s + 6·44-s + ⋯
L(s)  = 1  + 0.707·2-s − 3/2·4-s + 2.23·5-s + 1.88·7-s − 1.06·8-s + 1.58·10-s − 0.603·11-s + 1.38·13-s + 1.33·14-s + 3/4·16-s − 3.15·17-s − 3.35·20-s − 0.426·22-s − 0.417·23-s + 11/5·25-s + 0.980·26-s − 2.83·28-s + 0.928·29-s + 3.05·31-s + 0.707·32-s − 2.22·34-s + 4.22·35-s − 1.15·37-s − 2.37·40-s + 0.780·41-s + 0.152·43-s + 0.904·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 7^{5} \cdot 41^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 7^{5} \cdot 41^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{10} \cdot 7^{5} \cdot 41^{5}\)
Sign: $1$
Analytic conductor: \(3.73256\times 10^{6}\)
Root analytic conductor: \(4.54151\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 3^{10} \cdot 7^{5} \cdot 41^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.223421460\)
\(L(\frac12)\) \(\approx\) \(9.223421460\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
7$C_1$ \( ( 1 - T )^{5} \)
41$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - T + p^{2} T^{2} - p^{2} T^{3} + 5 p T^{4} - 11 T^{5} + 5 p^{2} T^{6} - p^{4} T^{7} + p^{5} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 - p T + 14 T^{2} - 14 T^{3} - 11 T^{4} + 86 T^{5} - 11 p T^{6} - 14 p^{2} T^{7} + 14 p^{3} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 2 T - 8 T^{2} - 52 T^{3} + 103 T^{4} + 844 T^{5} + 103 p T^{6} - 52 p^{2} T^{7} - 8 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 5 T + 56 T^{2} - 180 T^{3} + 1219 T^{4} - 2941 T^{5} + 1219 p T^{6} - 180 p^{2} T^{7} + 56 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 13 T + 110 T^{2} + 580 T^{3} + 2611 T^{4} + 10157 T^{5} + 2611 p T^{6} + 580 p^{2} T^{7} + 110 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 47 T^{2} - 132 T^{3} + 851 T^{4} - 5017 T^{5} + 851 p T^{6} - 132 p^{2} T^{7} + 47 p^{3} T^{8} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 2 T + 49 T^{2} + 158 T^{3} + 1885 T^{4} + 3835 T^{5} + 1885 p T^{6} + 158 p^{2} T^{7} + 49 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 5 T + 74 T^{2} - 230 T^{3} + 2629 T^{4} - 6442 T^{5} + 2629 p T^{6} - 230 p^{2} T^{7} + 74 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 17 T + 226 T^{2} - 2084 T^{3} + 16065 T^{4} - 96590 T^{5} + 16065 p T^{6} - 2084 p^{2} T^{7} + 226 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 7 T + 149 T^{2} + 879 T^{3} + 9772 T^{4} + 45884 T^{5} + 9772 p T^{6} + 879 p^{2} T^{7} + 149 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - T + 102 T^{2} - 198 T^{3} + 6135 T^{4} - 15081 T^{5} + 6135 p T^{6} - 198 p^{2} T^{7} + 102 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 9 T + 115 T^{2} + 653 T^{3} + 5488 T^{4} + 31712 T^{5} + 5488 p T^{6} + 653 p^{2} T^{7} + 115 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 5 T + 156 T^{2} + 36 T^{3} + 7939 T^{4} - 26602 T^{5} + 7939 p T^{6} + 36 p^{2} T^{7} + 156 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 7 T + 124 T^{2} + 772 T^{3} + 12043 T^{4} + 63362 T^{5} + 12043 p T^{6} + 772 p^{2} T^{7} + 124 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 22 T + 458 T^{2} - 5620 T^{3} + 64261 T^{4} - 519412 T^{5} + 64261 p T^{6} - 5620 p^{2} T^{7} + 458 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 3 T + 230 T^{2} + 228 T^{3} + 22853 T^{4} + 3146 T^{5} + 22853 p T^{6} + 228 p^{2} T^{7} + 230 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 24 T + 396 T^{2} - 4954 T^{3} + 55267 T^{4} - 504628 T^{5} + 55267 p T^{6} - 4954 p^{2} T^{7} + 396 p^{3} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 40 T + 950 T^{2} - 15562 T^{3} + 192937 T^{4} - 1857916 T^{5} + 192937 p T^{6} - 15562 p^{2} T^{7} + 950 p^{3} T^{8} - 40 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 42 T + 967 T^{2} + 15192 T^{3} + 184214 T^{4} + 1801084 T^{5} + 184214 p T^{6} + 15192 p^{2} T^{7} + 967 p^{3} T^{8} + 42 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 12 T + 156 T^{2} - 1442 T^{3} + 16075 T^{4} - 98732 T^{5} + 16075 p T^{6} - 1442 p^{2} T^{7} + 156 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 8 T + 457 T^{2} + 2846 T^{3} + 82405 T^{4} + 379849 T^{5} + 82405 p T^{6} + 2846 p^{2} T^{7} + 457 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 16 T + 323 T^{2} - 4394 T^{3} + 51905 T^{4} - 561841 T^{5} + 51905 p T^{6} - 4394 p^{2} T^{7} + 323 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.07929170631309858962992039037, −5.05356947271252682217065968634, −5.04516550219879289519868737073, −4.84036430484875676124238301840, −4.80721592155265660581996561574, −4.37963790506389762853610374524, −4.15862689384707400700842095255, −4.14240518196456688220343431743, −4.12998784895260265748326295629, −4.12263332380968395565829395498, −3.51568980803423939257259589171, −3.49440172688911640791885968725, −3.10154714541824482289137422632, −2.93634461950996874307253677363, −2.60069117858613596252637627864, −2.54145704620554990643730587048, −2.32425975236011734928205809944, −2.17226482455962211273589751446, −1.87836603040906460612751222704, −1.71206049816177495705416488967, −1.64109128544307211206890206013, −1.08044488103993797164343954508, −1.02876919361776193338675305527, −0.67432600640433240134757344861, −0.32437105844160595261655921017, 0.32437105844160595261655921017, 0.67432600640433240134757344861, 1.02876919361776193338675305527, 1.08044488103993797164343954508, 1.64109128544307211206890206013, 1.71206049816177495705416488967, 1.87836603040906460612751222704, 2.17226482455962211273589751446, 2.32425975236011734928205809944, 2.54145704620554990643730587048, 2.60069117858613596252637627864, 2.93634461950996874307253677363, 3.10154714541824482289137422632, 3.49440172688911640791885968725, 3.51568980803423939257259589171, 4.12263332380968395565829395498, 4.12998784895260265748326295629, 4.14240518196456688220343431743, 4.15862689384707400700842095255, 4.37963790506389762853610374524, 4.80721592155265660581996561574, 4.84036430484875676124238301840, 5.04516550219879289519868737073, 5.05356947271252682217065968634, 5.07929170631309858962992039037

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