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Properties

Label 10-5600e5-1.1-c1e5-0-2
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 $5$

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: \(5\)
Selberg data: \((10,\ 2^{25} \cdot 5^{10} \cdot 7^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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

−5.07028565899678709576492666259, −4.93614855603382731102351540567, −4.93014350589475383588822831675, −4.91639290719884951045171115226, −4.70555918111964619058807055066, −4.48999636102597240535102134385, −4.26023718397648838426687060653, −4.13922211381404490886017676058, −3.93627303314687884620196437015, −3.87794255077213806901441582935, −3.56500580394084158913334871240, −3.55231485200800665731509697042, −3.21038788333756225635105166452, −3.07111224294414042672775889838, −2.95234920274293456727437722950, −2.57792982013325235526421499091, −2.45236215799296554032595771147, −2.32777432638645061822205339675, −2.14618260705048781307037993878, −1.99061831330787287860444847177, −1.75325033692285621624812507674, −1.57232602543031378600609327308, −1.37698629071389629192202289356, −1.11933883000025312309779541770, −1.07569586094898301888645001506, 0, 0, 0, 0, 0, 1.07569586094898301888645001506, 1.11933883000025312309779541770, 1.37698629071389629192202289356, 1.57232602543031378600609327308, 1.75325033692285621624812507674, 1.99061831330787287860444847177, 2.14618260705048781307037993878, 2.32777432638645061822205339675, 2.45236215799296554032595771147, 2.57792982013325235526421499091, 2.95234920274293456727437722950, 3.07111224294414042672775889838, 3.21038788333756225635105166452, 3.55231485200800665731509697042, 3.56500580394084158913334871240, 3.87794255077213806901441582935, 3.93627303314687884620196437015, 4.13922211381404490886017676058, 4.26023718397648838426687060653, 4.48999636102597240535102134385, 4.70555918111964619058807055066, 4.91639290719884951045171115226, 4.93014350589475383588822831675, 4.93614855603382731102351540567, 5.07028565899678709576492666259

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