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Properties

Label 10-5600e5-1.1-c1e5-0-0
Degree $10$
Conductor $5.507\times 10^{18}$
Sign $1$
Analytic cond. $1.78782\times 10^{8}$
Root an. cond. $6.68701$
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 − 5·7-s − 2·9-s + 4·11-s − 4·17-s + 12·19-s − 10·21-s − 8·23-s − 4·27-s − 12·29-s + 12·31-s + 8·33-s − 12·37-s − 2·41-s + 8·43-s − 14·47-s + 15·49-s − 8·51-s − 8·53-s + 24·57-s + 16·59-s − 10·61-s + 10·63-s + 4·67-s − 16·69-s + 4·71-s + 12·73-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.88·7-s − 2/3·9-s + 1.20·11-s − 0.970·17-s + 2.75·19-s − 2.18·21-s − 1.66·23-s − 0.769·27-s − 2.22·29-s + 2.15·31-s + 1.39·33-s − 1.97·37-s − 0.312·41-s + 1.21·43-s − 2.04·47-s + 15/7·49-s − 1.12·51-s − 1.09·53-s + 3.17·57-s + 2.08·59-s − 1.28·61-s + 1.25·63-s + 0.488·67-s − 1.92·69-s + 0.474·71-s + 1.40·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{25} \cdot 5^{10} \cdot 7^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{25} \cdot 5^{10} \cdot 7^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{25} \cdot 5^{10} \cdot 7^{5}\)
Sign: $1$
Analytic conductor: \(1.78782\times 10^{8}\)
Root analytic conductor: \(6.68701\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{25} \cdot 5^{10} \cdot 7^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(8.465700789\)
\(L(\frac12)\) \(\approx\) \(8.465700789\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7$C_1$ \( ( 1 + T )^{5} \)
good3$C_2 \wr S_5$ \( 1 - 2 T + 2 p T^{2} - 4 p T^{3} + p^{3} T^{4} - 52 T^{5} + p^{4} T^{6} - 4 p^{3} T^{7} + 2 p^{4} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 4 T + 30 T^{2} - 24 T^{3} + 137 T^{4} + 568 T^{5} + 137 p T^{6} - 24 p^{2} T^{7} + 30 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 18 T^{2} - 80 T^{3} + 235 T^{4} - 1256 T^{5} + 235 p T^{6} - 80 p^{2} T^{7} + 18 p^{3} T^{8} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 4 T + 60 T^{2} + 252 T^{3} + 1691 T^{4} + 6224 T^{5} + 1691 p T^{6} + 252 p^{2} T^{7} + 60 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 12 T + 125 T^{2} - 840 T^{3} + 5136 T^{4} - 23512 T^{5} + 5136 p T^{6} - 840 p^{2} T^{7} + 125 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 8 T + 95 T^{2} + 592 T^{3} + 3974 T^{4} + 19280 T^{5} + 3974 p T^{6} + 592 p^{2} T^{7} + 95 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 12 T + 130 T^{2} + 930 T^{3} + 6981 T^{4} + 38500 T^{5} + 6981 p T^{6} + 930 p^{2} T^{7} + 130 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 12 T + 99 T^{2} - 304 T^{3} - 78 T^{4} + 8312 T^{5} - 78 p T^{6} - 304 p^{2} T^{7} + 99 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 12 T + 189 T^{2} + 1520 T^{3} + 13878 T^{4} + 79880 T^{5} + 13878 p T^{6} + 1520 p^{2} T^{7} + 189 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 2 T + 105 T^{2} - 304 T^{3} + 3454 T^{4} - 31780 T^{5} + 3454 p T^{6} - 304 p^{2} T^{7} + 105 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 8 T + 75 T^{2} - 640 T^{3} + 5262 T^{4} - 40560 T^{5} + 5262 p T^{6} - 640 p^{2} T^{7} + 75 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 14 T + 272 T^{2} + 2472 T^{3} + 26691 T^{4} + 170452 T^{5} + 26691 p T^{6} + 2472 p^{2} T^{7} + 272 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 8 T + 193 T^{2} + 1200 T^{3} + 17042 T^{4} + 84112 T^{5} + 17042 p T^{6} + 1200 p^{2} T^{7} + 193 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 16 T + 189 T^{2} - 1728 T^{3} + 18280 T^{4} - 150624 T^{5} + 18280 p T^{6} - 1728 p^{2} T^{7} + 189 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 10 T + 259 T^{2} + 1972 T^{3} + 28864 T^{4} + 167812 T^{5} + 28864 p T^{6} + 1972 p^{2} T^{7} + 259 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 4 T + 227 T^{2} - 1264 T^{3} + 23502 T^{4} - 132952 T^{5} + 23502 p T^{6} - 1264 p^{2} T^{7} + 227 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 4 T + 127 T^{2} + 208 T^{3} - 394 T^{4} + 70888 T^{5} - 394 p T^{6} + 208 p^{2} T^{7} + 127 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 12 T + 325 T^{2} - 3008 T^{3} + 44674 T^{4} - 313320 T^{5} + 44674 p T^{6} - 3008 p^{2} T^{7} + 325 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 32 T + 758 T^{2} - 11996 T^{3} + 152941 T^{4} - 1499848 T^{5} + 152941 p T^{6} - 11996 p^{2} T^{7} + 758 p^{3} T^{8} - 32 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 24 T + 361 T^{2} - 3936 T^{3} + 35124 T^{4} - 295312 T^{5} + 35124 p T^{6} - 3936 p^{2} T^{7} + 361 p^{3} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 2 T + 277 T^{2} - 312 T^{3} + 36290 T^{4} - 26956 T^{5} + 36290 p T^{6} - 312 p^{2} T^{7} + 277 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 12 T + 420 T^{2} - 3500 T^{3} + 72531 T^{4} - 456608 T^{5} + 72531 p T^{6} - 3500 p^{2} T^{7} + 420 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.86971940120555707221328814591, −4.50574186261911905798069392165, −4.48776570592177616794813360708, −4.15533084355264114664791497034, −4.01229959374197329177559996445, −3.99353098640246013194934149256, −3.73681776039313803241198675632, −3.53328762340042516021577283090, −3.52651083117213592123158948599, −3.41463386972438111615781840360, −3.10054224256617794106868279018, −2.99864774870946295820585247435, −2.89149734171179012868462442999, −2.70337267959605891338894599930, −2.69667887694498289936877125006, −2.16411705462746302594051441886, −2.10278140699529801088120842152, −1.84200403894685982849008271958, −1.80034347335116663175778603044, −1.69884704960848889969984784254, −1.24929459200713939090622463434, −0.845668371282729418860855212224, −0.62246974918979301490794540836, −0.54524751564144778872751004219, −0.35409550608324087215259992926, 0.35409550608324087215259992926, 0.54524751564144778872751004219, 0.62246974918979301490794540836, 0.845668371282729418860855212224, 1.24929459200713939090622463434, 1.69884704960848889969984784254, 1.80034347335116663175778603044, 1.84200403894685982849008271958, 2.10278140699529801088120842152, 2.16411705462746302594051441886, 2.69667887694498289936877125006, 2.70337267959605891338894599930, 2.89149734171179012868462442999, 2.99864774870946295820585247435, 3.10054224256617794106868279018, 3.41463386972438111615781840360, 3.52651083117213592123158948599, 3.53328762340042516021577283090, 3.73681776039313803241198675632, 3.99353098640246013194934149256, 4.01229959374197329177559996445, 4.15533084355264114664791497034, 4.48776570592177616794813360708, 4.50574186261911905798069392165, 4.86971940120555707221328814591

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