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Properties

Label 10-9522e5-1.1-c1e5-0-7
Degree $10$
Conductor $7.828\times 10^{19}$
Sign $-1$
Analytic cond. $2.54112\times 10^{9}$
Root an. cond. $8.71972$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $5$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s + 15·4-s − 7·5-s + 7·7-s − 35·8-s + 35·10-s − 13·11-s − 4·13-s − 35·14-s + 70·16-s − 9·17-s + 11·19-s − 105·20-s + 65·22-s + 11·25-s + 20·26-s + 105·28-s + 7·29-s − 8·31-s − 126·32-s + 45·34-s − 49·35-s + 12·37-s − 55·38-s + 245·40-s + 10·41-s + 4·43-s + ⋯
L(s)  = 1  − 3.53·2-s + 15/2·4-s − 3.13·5-s + 2.64·7-s − 12.3·8-s + 11.0·10-s − 3.91·11-s − 1.10·13-s − 9.35·14-s + 35/2·16-s − 2.18·17-s + 2.52·19-s − 23.4·20-s + 13.8·22-s + 11/5·25-s + 3.92·26-s + 19.8·28-s + 1.29·29-s − 1.43·31-s − 22.2·32-s + 7.71·34-s − 8.28·35-s + 1.97·37-s − 8.92·38-s + 38.7·40-s + 1.56·41-s + 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{5} \cdot 3^{10} \cdot 23^{10}\)
Sign: $-1$
Analytic conductor: \(2.54112\times 10^{9}\)
Root analytic conductor: \(8.71972\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{5} \cdot 3^{10} \cdot 23^{10} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 + T )^{5} \)
3 \( 1 \)
23 \( 1 \)
good5$C_2 \times (C_2^4 : C_5)$ \( 1 + 7 T + 38 T^{2} + 134 T^{3} + 82 p T^{4} + 967 T^{5} + 82 p^{2} T^{6} + 134 p^{2} T^{7} + 38 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \times (C_2^4 : C_5)$ \( 1 - p T + 48 T^{2} - 190 T^{3} + 104 p T^{4} - 1951 T^{5} + 104 p^{2} T^{6} - 190 p^{2} T^{7} + 48 p^{3} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
11$C_2 \times (C_2^4 : C_5)$ \( 1 + 13 T + 105 T^{2} + 625 T^{3} + 2875 T^{4} + 10603 T^{5} + 2875 p T^{6} + 625 p^{2} T^{7} + 105 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \times (C_2^4 : C_5)$ \( 1 + 4 T + 56 T^{2} + 181 T^{3} + 102 p T^{4} + 3355 T^{5} + 102 p^{2} T^{6} + 181 p^{2} T^{7} + 56 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \times (C_2^4 : C_5)$ \( 1 + 9 T + 6 p T^{2} + 581 T^{3} + 3673 T^{4} + 14509 T^{5} + 3673 p T^{6} + 581 p^{2} T^{7} + 6 p^{4} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \times (C_2^4 : C_5)$ \( 1 - 11 T + 106 T^{2} - 660 T^{3} + 3874 T^{4} - 17391 T^{5} + 3874 p T^{6} - 660 p^{2} T^{7} + 106 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \times (C_2^4 : C_5)$ \( 1 - 7 T + 70 T^{2} - 399 T^{3} + 3049 T^{4} - 13061 T^{5} + 3049 p T^{6} - 399 p^{2} T^{7} + 70 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \times (C_2^4 : C_5)$ \( 1 + 8 T + 86 T^{2} + 259 T^{3} + 1632 T^{4} - 265 T^{5} + 1632 p T^{6} + 259 p^{2} T^{7} + 86 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \times (C_2^4 : C_5)$ \( 1 - 12 T + 236 T^{2} - 1872 T^{3} + 19431 T^{4} - 105695 T^{5} + 19431 p T^{6} - 1872 p^{2} T^{7} + 236 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \times (C_2^4 : C_5)$ \( 1 - 10 T + 168 T^{2} - 1445 T^{3} + 12064 T^{4} - 84847 T^{5} + 12064 p T^{6} - 1445 p^{2} T^{7} + 168 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \times (C_2^4 : C_5)$ \( 1 - 4 T + 118 T^{2} - 837 T^{3} + 6250 T^{4} - 56751 T^{5} + 6250 p T^{6} - 837 p^{2} T^{7} + 118 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \times (C_2^4 : C_5)$ \( 1 - 24 T + 318 T^{2} - 2992 T^{3} + 23413 T^{4} - 164567 T^{5} + 23413 p T^{6} - 2992 p^{2} T^{7} + 318 p^{3} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \times (C_2^4 : C_5)$ \( 1 + 9 T + 95 T^{2} + 458 T^{3} + 4749 T^{4} + 16995 T^{5} + 4749 p T^{6} + 458 p^{2} T^{7} + 95 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \times (C_2^4 : C_5)$ \( 1 - 14 T + 292 T^{2} - 2772 T^{3} + 33895 T^{4} - 235261 T^{5} + 33895 p T^{6} - 2772 p^{2} T^{7} + 292 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \times (C_2^4 : C_5)$ \( 1 - 5 T + 62 T^{2} - 262 T^{3} + 5022 T^{4} - 47687 T^{5} + 5022 p T^{6} - 262 p^{2} T^{7} + 62 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \times (C_2^4 : C_5)$ \( 1 - 13 T + 154 T^{2} - 1359 T^{3} + 10801 T^{4} - 107873 T^{5} + 10801 p T^{6} - 1359 p^{2} T^{7} + 154 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \times (C_2^4 : C_5)$ \( 1 - 19 T + 319 T^{2} - 3717 T^{3} + 42237 T^{4} - 374865 T^{5} + 42237 p T^{6} - 3717 p^{2} T^{7} + 319 p^{3} T^{8} - 19 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \times (C_2^4 : C_5)$ \( 1 - 4 T + 48 T^{2} - 569 T^{3} + 8868 T^{4} - 25065 T^{5} + 8868 p T^{6} - 569 p^{2} T^{7} + 48 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \times (C_2^4 : C_5)$ \( 1 - 4 T + 144 T^{2} - 1149 T^{3} + 19850 T^{4} - 82445 T^{5} + 19850 p T^{6} - 1149 p^{2} T^{7} + 144 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \times (C_2^4 : C_5)$ \( 1 + 24 T + 465 T^{2} + 6217 T^{3} + 75404 T^{4} + 727023 T^{5} + 75404 p T^{6} + 6217 p^{2} T^{7} + 465 p^{3} T^{8} + 24 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \times (C_2^4 : C_5)$ \( 1 + 4 T + 315 T^{2} + 627 T^{3} + 45554 T^{4} + 57875 T^{5} + 45554 p T^{6} + 627 p^{2} T^{7} + 315 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \times (C_2^4 : C_5)$ \( 1 + 9 T + 150 T^{2} + 150 T^{3} + 11624 T^{4} + 9119 T^{5} + 11624 p T^{6} + 150 p^{2} T^{7} + 150 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.96957207049575090991567535287, −4.65752146812826728810398589017, −4.49965992194647599791260264974, −4.49755859300161687656040265245, −4.37192816660276376440492046060, −4.01552624374943785312226106544, −3.83993891266461238603917901593, −3.74793563385811383306121862179, −3.71841849978914328640501628349, −3.68072151817686056945872785138, −2.99700896102542309198228332799, −2.94853616086692601302623598346, −2.92315889782461262007167468131, −2.79402458429724765010869675784, −2.56680216121426342488930822294, −2.27217255458666217884605700587, −2.25346464943294169877895589905, −2.22191929488478954494269398869, −2.06026555809045066942110167946, −1.93430312335768911750720636634, −1.23461609270185795328776236053, −1.22035675696718519935120637470, −1.11926976512781448133536312908, −0.971164399633663490679782312942, −0.812367662962477066641049934824, 0, 0, 0, 0, 0, 0.812367662962477066641049934824, 0.971164399633663490679782312942, 1.11926976512781448133536312908, 1.22035675696718519935120637470, 1.23461609270185795328776236053, 1.93430312335768911750720636634, 2.06026555809045066942110167946, 2.22191929488478954494269398869, 2.25346464943294169877895589905, 2.27217255458666217884605700587, 2.56680216121426342488930822294, 2.79402458429724765010869675784, 2.92315889782461262007167468131, 2.94853616086692601302623598346, 2.99700896102542309198228332799, 3.68072151817686056945872785138, 3.71841849978914328640501628349, 3.74793563385811383306121862179, 3.83993891266461238603917901593, 4.01552624374943785312226106544, 4.37192816660276376440492046060, 4.49755859300161687656040265245, 4.49965992194647599791260264974, 4.65752146812826728810398589017, 4.96957207049575090991567535287

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