Properties

Label 10-2842e5-1.1-c1e5-0-6
Degree $10$
Conductor $1.854\times 10^{17}$
Sign $-1$
Analytic cond. $6.01874\times 10^{6}$
Root an. cond. $4.76376$
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 − 3·3-s + 15·4-s − 7·5-s − 15·6-s + 35·8-s − 9-s − 35·10-s − 45·12-s − 8·13-s + 21·15-s + 70·16-s − 16·17-s − 5·18-s − 2·19-s − 105·20-s − 5·23-s − 105·24-s + 14·25-s − 40·26-s + 12·27-s + 5·29-s + 105·30-s − 5·31-s + 126·32-s − 80·34-s − 15·36-s + ⋯
L(s)  = 1  + 3.53·2-s − 1.73·3-s + 15/2·4-s − 3.13·5-s − 6.12·6-s + 12.3·8-s − 1/3·9-s − 11.0·10-s − 12.9·12-s − 2.21·13-s + 5.42·15-s + 35/2·16-s − 3.88·17-s − 1.17·18-s − 0.458·19-s − 23.4·20-s − 1.04·23-s − 21.4·24-s + 14/5·25-s − 7.84·26-s + 2.30·27-s + 0.928·29-s + 19.1·30-s − 0.898·31-s + 22.2·32-s − 13.7·34-s − 5/2·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 7^{10} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 7^{10} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{5} \cdot 7^{10} \cdot 29^{5}\)
Sign: $-1$
Analytic conductor: \(6.01874\times 10^{6}\)
Root analytic conductor: \(4.76376\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{5} \cdot 7^{10} \cdot 29^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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$C_1$ \( ( 1 - T )^{5} \)
7 \( 1 \)
29$C_1$ \( ( 1 - T )^{5} \)
good3$C_2 \wr S_5$ \( 1 + p T + 10 T^{2} + 7 p T^{3} + 17 p T^{4} + 83 T^{5} + 17 p^{2} T^{6} + 7 p^{3} T^{7} + 10 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 7 T + 7 p T^{2} + 131 T^{3} + 79 p T^{4} + 963 T^{5} + 79 p^{2} T^{6} + 131 p^{2} T^{7} + 7 p^{4} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 26 T^{2} + 60 T^{3} + 271 T^{4} + 1259 T^{5} + 271 p T^{6} + 60 p^{2} T^{7} + 26 p^{3} T^{8} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 8 T + 55 T^{2} + 187 T^{3} + 660 T^{4} + 1619 T^{5} + 660 p T^{6} + 187 p^{2} T^{7} + 55 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 16 T + 152 T^{2} + 1002 T^{3} + 5271 T^{4} + 23289 T^{5} + 5271 p T^{6} + 1002 p^{2} T^{7} + 152 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 2 T + 58 T^{2} + 170 T^{3} + 1673 T^{4} + 4895 T^{5} + 1673 p T^{6} + 170 p^{2} T^{7} + 58 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 5 T + 86 T^{2} + 433 T^{3} + 3415 T^{4} + 14579 T^{5} + 3415 p T^{6} + 433 p^{2} T^{7} + 86 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 5 T + 67 T^{2} + 431 T^{3} + 3641 T^{4} + 13847 T^{5} + 3641 p T^{6} + 431 p^{2} T^{7} + 67 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 164 T^{2} + 26 T^{3} + 11355 T^{4} + 1921 T^{5} + 11355 p T^{6} + 26 p^{2} T^{7} + 164 p^{3} T^{8} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 17 T + 273 T^{2} + 2525 T^{3} + 22833 T^{4} + 145759 T^{5} + 22833 p T^{6} + 2525 p^{2} T^{7} + 273 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - T + 198 T^{2} - 161 T^{3} + 16367 T^{4} - 10163 T^{5} + 16367 p T^{6} - 161 p^{2} T^{7} + 198 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 4 T + 132 T^{2} + 20 T^{3} + 6503 T^{4} + 17815 T^{5} + 6503 p T^{6} + 20 p^{2} T^{7} + 132 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 5 T + 30 T^{2} - 935 T^{3} + 5975 T^{4} - 20395 T^{5} + 5975 p T^{6} - 935 p^{2} T^{7} + 30 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 17 T + 342 T^{2} + 3635 T^{3} + 42009 T^{4} + 312781 T^{5} + 42009 p T^{6} + 3635 p^{2} T^{7} + 342 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 13 T + 295 T^{2} + 3042 T^{3} + 35261 T^{4} + 274399 T^{5} + 35261 p T^{6} + 3042 p^{2} T^{7} + 295 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 14 T + 271 T^{2} + 2041 T^{3} + 23868 T^{4} + 135911 T^{5} + 23868 p T^{6} + 2041 p^{2} T^{7} + 271 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 8 T + 248 T^{2} + 2348 T^{3} + 27969 T^{4} + 252795 T^{5} + 27969 p T^{6} + 2348 p^{2} T^{7} + 248 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 6 T + 182 T^{2} + 1486 T^{3} + 19017 T^{4} + 159737 T^{5} + 19017 p T^{6} + 1486 p^{2} T^{7} + 182 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 11 T + 138 T^{2} - 367 T^{3} - 267 T^{4} + 79087 T^{5} - 267 p T^{6} - 367 p^{2} T^{7} + 138 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 2 T + 122 T^{2} - 295 T^{3} + 6762 T^{4} - 65207 T^{5} + 6762 p T^{6} - 295 p^{2} T^{7} + 122 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 9 T + 188 T^{2} + 2628 T^{3} + 29254 T^{4} + 273073 T^{5} + 29254 p T^{6} + 2628 p^{2} T^{7} + 188 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 12 T + 210 T^{2} + 252 T^{3} + 5347 T^{4} - 118835 T^{5} + 5347 p T^{6} + 252 p^{2} T^{7} + 210 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.39776759759863254610066475449, −5.34869432723042046199494224802, −5.23799654308104749375272287482, −5.22590616590932448791494252685, −4.90924106900620703408684694946, −4.72427817824212049332715052413, −4.53403784796927866451706695307, −4.48232370437648394249605791105, −4.42465933335502534457355633561, −4.23908326502254540842305609327, −3.93119631014163179451695853787, −3.91636392778402160902060637067, −3.72240404407528417074867975866, −3.69777741980674988200530810567, −3.45597454076824555006239620412, −2.98199723886332156165486614855, −2.79519151764321445708574017486, −2.77855420524218318887187999432, −2.61705727674035541411957522853, −2.55531640420683251630748490223, −2.15170747125300372431730710450, −1.99504654581788766271095165655, −1.63151978422710404275866692132, −1.46422567404713019954794541598, −1.17480409409292014111448524397, 0, 0, 0, 0, 0, 1.17480409409292014111448524397, 1.46422567404713019954794541598, 1.63151978422710404275866692132, 1.99504654581788766271095165655, 2.15170747125300372431730710450, 2.55531640420683251630748490223, 2.61705727674035541411957522853, 2.77855420524218318887187999432, 2.79519151764321445708574017486, 2.98199723886332156165486614855, 3.45597454076824555006239620412, 3.69777741980674988200530810567, 3.72240404407528417074867975866, 3.91636392778402160902060637067, 3.93119631014163179451695853787, 4.23908326502254540842305609327, 4.42465933335502534457355633561, 4.48232370437648394249605791105, 4.53403784796927866451706695307, 4.72427817824212049332715052413, 4.90924106900620703408684694946, 5.22590616590932448791494252685, 5.23799654308104749375272287482, 5.34869432723042046199494224802, 5.39776759759863254610066475449

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