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Properties

Label 10-4235e5-1.1-c1e5-0-0
Degree $10$
Conductor $1.362\times 10^{18}$
Sign $1$
Analytic cond. $4.42234\times 10^{7}$
Root an. cond. $5.81520$
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  − 2·3-s − 3·4-s − 5·5-s − 5·7-s + 4·8-s + 6·12-s + 10·13-s + 10·15-s + 5·16-s + 2·17-s − 3·19-s + 15·20-s + 10·21-s + 3·23-s − 8·24-s + 15·25-s + 27-s + 15·28-s + 29-s − 12·31-s − 11·32-s + 25·35-s − 20·39-s − 20·40-s − 11·41-s + 10·43-s + 16·47-s + ⋯
L(s)  = 1  − 1.15·3-s − 3/2·4-s − 2.23·5-s − 1.88·7-s + 1.41·8-s + 1.73·12-s + 2.77·13-s + 2.58·15-s + 5/4·16-s + 0.485·17-s − 0.688·19-s + 3.35·20-s + 2.18·21-s + 0.625·23-s − 1.63·24-s + 3·25-s + 0.192·27-s + 2.83·28-s + 0.185·29-s − 2.15·31-s − 1.94·32-s + 4.22·35-s − 3.20·39-s − 3.16·40-s − 1.71·41-s + 1.52·43-s + 2.33·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 7^{5} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 7^{5} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(5^{5} \cdot 7^{5} \cdot 11^{10}\)
Sign: $1$
Analytic conductor: \(4.42234\times 10^{7}\)
Root analytic conductor: \(5.81520\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 5^{5} \cdot 7^{5} \cdot 11^{10} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.212653002\)
\(L(\frac12)\) \(\approx\) \(1.212653002\)
\(L(\frac{3}{2})\) 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)$
bad5$C_1$ \( ( 1 + T )^{5} \)
7$C_1$ \( ( 1 + T )^{5} \)
11 \( 1 \)
good2$C_2 \wr S_5$ \( 1 + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} - 13 T^{5} + p^{3} T^{6} - p^{4} T^{7} + 3 p^{3} T^{8} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 + 2 T + 4 T^{2} + 7 T^{3} + 2 p^{2} T^{4} + 25 T^{5} + 2 p^{3} T^{6} + 7 p^{2} T^{7} + 4 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 10 T + 73 T^{2} - 421 T^{3} + 1998 T^{4} - 7699 T^{5} + 1998 p T^{6} - 421 p^{2} T^{7} + 73 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 2 T + 57 T^{2} - 93 T^{3} + 1546 T^{4} - 1997 T^{5} + 1546 p T^{6} - 93 p^{2} T^{7} + 57 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 3 T + 44 T^{2} + 6 p T^{3} + 1142 T^{4} + 1897 T^{5} + 1142 p T^{6} + 6 p^{3} T^{7} + 44 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 3 T + 45 T^{2} - 67 T^{3} + 1651 T^{4} - 3481 T^{5} + 1651 p T^{6} - 67 p^{2} T^{7} + 45 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - T + 109 T^{2} - T^{3} + 5251 T^{4} + 1477 T^{5} + 5251 p T^{6} - p^{2} T^{7} + 109 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 12 T + 116 T^{2} + 453 T^{3} + 1616 T^{4} - 139 T^{5} + 1616 p T^{6} + 453 p^{2} T^{7} + 116 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 40 T^{2} - 243 T^{3} + 2144 T^{4} - 6437 T^{5} + 2144 p T^{6} - 243 p^{2} T^{7} + 40 p^{3} T^{8} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 11 T + 175 T^{2} + 1223 T^{3} + 11437 T^{4} + 61879 T^{5} + 11437 p T^{6} + 1223 p^{2} T^{7} + 175 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 10 T + 218 T^{2} - 1473 T^{3} + 18040 T^{4} - 88937 T^{5} + 18040 p T^{6} - 1473 p^{2} T^{7} + 218 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 16 T + 256 T^{2} - 2458 T^{3} + 23647 T^{4} - 161699 T^{5} + 23647 p T^{6} - 2458 p^{2} T^{7} + 256 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 4 T + 98 T^{2} - 193 T^{3} + 4696 T^{4} - 293 T^{5} + 4696 p T^{6} - 193 p^{2} T^{7} + 98 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 32 T + 675 T^{2} + 9597 T^{3} + 106822 T^{4} + 912929 T^{5} + 106822 p T^{6} + 9597 p^{2} T^{7} + 675 p^{3} T^{8} + 32 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 40 T + 918 T^{2} - 14160 T^{3} + 163483 T^{4} - 1443777 T^{5} + 163483 p T^{6} - 14160 p^{2} T^{7} + 918 p^{3} T^{8} - 40 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 7 T + 218 T^{2} - 1484 T^{3} + 24806 T^{4} - 136015 T^{5} + 24806 p T^{6} - 1484 p^{2} T^{7} + 218 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 10 T + 320 T^{2} - 2293 T^{3} + 42034 T^{4} - 225941 T^{5} + 42034 p T^{6} - 2293 p^{2} T^{7} + 320 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 11 T + 224 T^{2} - 2294 T^{3} + 23764 T^{4} - 217565 T^{5} + 23764 p T^{6} - 2294 p^{2} T^{7} + 224 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 13 T + 300 T^{2} - 2988 T^{3} + 40636 T^{4} - 309897 T^{5} + 40636 p T^{6} - 2988 p^{2} T^{7} + 300 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 26 T + 531 T^{2} - 6951 T^{3} + 83026 T^{4} - 772193 T^{5} + 83026 p T^{6} - 6951 p^{2} T^{7} + 531 p^{3} T^{8} - 26 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 5 T + 350 T^{2} - 1403 T^{3} + 55471 T^{4} - 171451 T^{5} + 55471 p T^{6} - 1403 p^{2} T^{7} + 350 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 5 T + 362 T^{2} + 1189 T^{3} + 58577 T^{4} + 136625 T^{5} + 58577 p T^{6} + 1189 p^{2} T^{7} + 362 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.81580139991474995116293291775, −4.81399139597817174136231920660, −4.59510634899489943845319566462, −4.54441049354589928428628165588, −4.19362077535801703880106367245, −3.97642233641425656363763708909, −3.94802612191560778657429263604, −3.89917753734479293685986716280, −3.74531366415385146452442066776, −3.56853038361322631736964548235, −3.55364622560951794528543635160, −3.25873849718233526684967533130, −3.05135857026083790753369647784, −2.96303546833957732912359404280, −2.65305383388792748122705376201, −2.37105861352619061696230374896, −2.10353132311963614335838832748, −1.82909213336936799541082374939, −1.79339357021378853339191389540, −1.31467197036473409063575310592, −0.941644749399451622078077557137, −0.864666209158323276601666248789, −0.72245090262618010498437483804, −0.44278343582204333048731020302, −0.30956054673123242424224800689, 0.30956054673123242424224800689, 0.44278343582204333048731020302, 0.72245090262618010498437483804, 0.864666209158323276601666248789, 0.941644749399451622078077557137, 1.31467197036473409063575310592, 1.79339357021378853339191389540, 1.82909213336936799541082374939, 2.10353132311963614335838832748, 2.37105861352619061696230374896, 2.65305383388792748122705376201, 2.96303546833957732912359404280, 3.05135857026083790753369647784, 3.25873849718233526684967533130, 3.55364622560951794528543635160, 3.56853038361322631736964548235, 3.74531366415385146452442066776, 3.89917753734479293685986716280, 3.94802612191560778657429263604, 3.97642233641425656363763708909, 4.19362077535801703880106367245, 4.54441049354589928428628165588, 4.59510634899489943845319566462, 4.81399139597817174136231920660, 4.81580139991474995116293291775

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