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Properties

Label 10-5600e5-1.1-c1e5-0-3
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

−5.33429487455412625229204301836, −4.82059331541477266671869783867, −4.80724287282601440974215470694, −4.76536092614231466464174309950, −4.63929748540847904185461433113, −4.33964593745352877949181849704, −3.96897632519477544810842159333, −3.94272715420831801876004802385, −3.92366034098256366230277375168, −3.89938906260111730192303773871, −3.53366750613250421050687685871, −3.33150992490162346551578295139, −3.21602528096057528992137000220, −3.07987805402367387224515842660, −3.04490667825277182659391323882, −2.62904919411856618565554554484, −2.39959973993398748485649891058, −2.39764273687916113850545161556, −2.38868025903287156558025875675, −2.22542064752122849660326372165, −1.90169512703764612182025121623, −1.53797523420719649800405955271, −1.44045951880787033607258509247, −1.09041915192200348143472168290, −1.08050327100651958044380615074, 0, 0, 0, 0, 0, 1.08050327100651958044380615074, 1.09041915192200348143472168290, 1.44045951880787033607258509247, 1.53797523420719649800405955271, 1.90169512703764612182025121623, 2.22542064752122849660326372165, 2.38868025903287156558025875675, 2.39764273687916113850545161556, 2.39959973993398748485649891058, 2.62904919411856618565554554484, 3.04490667825277182659391323882, 3.07987805402367387224515842660, 3.21602528096057528992137000220, 3.33150992490162346551578295139, 3.53366750613250421050687685871, 3.89938906260111730192303773871, 3.92366034098256366230277375168, 3.94272715420831801876004802385, 3.96897632519477544810842159333, 4.33964593745352877949181849704, 4.63929748540847904185461433113, 4.76536092614231466464174309950, 4.80724287282601440974215470694, 4.82059331541477266671869783867, 5.33429487455412625229204301836

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