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Properties

Label 10-6025e5-1.1-c1e5-0-0
Degree $10$
Conductor $7.939\times 10^{18}$
Sign $1$
Analytic cond. $2.57733\times 10^{8}$
Root an. cond. $6.93612$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 5·3-s − 3·4-s + 5·6-s + 10·7-s − 4·8-s + 8·9-s − 3·11-s − 15·12-s + 13-s + 10·14-s + 6·16-s + 5·17-s + 8·18-s + 3·19-s + 50·21-s − 3·22-s + 8·23-s − 20·24-s + 26-s − 2·27-s − 30·28-s + 9·29-s − 16·31-s + 13·32-s − 15·33-s + 5·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 2.88·3-s − 3/2·4-s + 2.04·6-s + 3.77·7-s − 1.41·8-s + 8/3·9-s − 0.904·11-s − 4.33·12-s + 0.277·13-s + 2.67·14-s + 3/2·16-s + 1.21·17-s + 1.88·18-s + 0.688·19-s + 10.9·21-s − 0.639·22-s + 1.66·23-s − 4.08·24-s + 0.196·26-s − 0.384·27-s − 5.66·28-s + 1.67·29-s − 2.87·31-s + 2.29·32-s − 2.61·33-s + 0.857·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 241^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 241^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(5^{10} \cdot 241^{5}\)
Sign: $1$
Analytic conductor: \(2.57733\times 10^{8}\)
Root analytic conductor: \(6.93612\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 5^{10} \cdot 241^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(98.32234376\)
\(L(\frac12)\) \(\approx\) \(98.32234376\)
\(L(\frac{3}{2})\) 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 \( 1 \)
241$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - T + p^{2} T^{2} - 3 T^{3} + 5 T^{4} - 5 T^{5} + 5 p T^{6} - 3 p^{2} T^{7} + p^{5} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 - 5 T + 17 T^{2} - 43 T^{3} + 11 p^{2} T^{4} - 185 T^{5} + 11 p^{3} T^{6} - 43 p^{2} T^{7} + 17 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 10 T + 9 p T^{2} - 277 T^{3} + 20 p^{2} T^{4} - 2809 T^{5} + 20 p^{3} T^{6} - 277 p^{2} T^{7} + 9 p^{4} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 3 T + 30 T^{2} + 64 T^{3} + 402 T^{4} + 681 T^{5} + 402 p T^{6} + 64 p^{2} T^{7} + 30 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - T + 48 T^{2} - 34 T^{3} + 1070 T^{4} - 547 T^{5} + 1070 p T^{6} - 34 p^{2} T^{7} + 48 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 5 T + 33 T^{2} - 181 T^{3} + 789 T^{4} - 3625 T^{5} + 789 p T^{6} - 181 p^{2} T^{7} + 33 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 3 T + 72 T^{2} - 170 T^{3} + 2368 T^{4} - 4385 T^{5} + 2368 p T^{6} - 170 p^{2} T^{7} + 72 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 8 T + 105 T^{2} - 615 T^{3} + 4696 T^{4} - 20085 T^{5} + 4696 p T^{6} - 615 p^{2} T^{7} + 105 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 9 T + 138 T^{2} - 782 T^{3} + 7068 T^{4} - 29739 T^{5} + 7068 p T^{6} - 782 p^{2} T^{7} + 138 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 16 T + 230 T^{2} + 2063 T^{3} + 16548 T^{4} + 97141 T^{5} + 16548 p T^{6} + 2063 p^{2} T^{7} + 230 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 7 T + 136 T^{2} + 19 p T^{3} + 8869 T^{4} + 36613 T^{5} + 8869 p T^{6} + 19 p^{3} T^{7} + 136 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 9 T + 232 T^{2} - 1503 T^{3} + 20127 T^{4} - 92977 T^{5} + 20127 p T^{6} - 1503 p^{2} T^{7} + 232 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 32 T + 556 T^{2} - 6781 T^{3} + 63014 T^{4} - 461877 T^{5} + 63014 p T^{6} - 6781 p^{2} T^{7} + 556 p^{3} T^{8} - 32 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 7 T + 210 T^{2} - 1139 T^{3} + 18579 T^{4} - 76733 T^{5} + 18579 p T^{6} - 1139 p^{2} T^{7} + 210 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 32 T + 655 T^{2} - 9013 T^{3} + 95970 T^{4} - 781289 T^{5} + 95970 p T^{6} - 9013 p^{2} T^{7} + 655 p^{3} T^{8} - 32 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 8 T + 291 T^{2} + 1707 T^{3} + 33590 T^{4} + 145309 T^{5} + 33590 p T^{6} + 1707 p^{2} T^{7} + 291 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 12 T + 226 T^{2} + 1909 T^{3} + 24382 T^{4} + 164073 T^{5} + 24382 p T^{6} + 1909 p^{2} T^{7} + 226 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 5 T + 152 T^{2} - 454 T^{3} + 14354 T^{4} - 39039 T^{5} + 14354 p T^{6} - 454 p^{2} T^{7} + 152 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 11 T + 293 T^{2} + 2666 T^{3} + 38033 T^{4} + 267381 T^{5} + 38033 p T^{6} + 2666 p^{2} T^{7} + 293 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 29 T + 676 T^{2} - 9972 T^{3} + 124570 T^{4} - 1149081 T^{5} + 124570 p T^{6} - 9972 p^{2} T^{7} + 676 p^{3} T^{8} - 29 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 16 T + 424 T^{2} - 4911 T^{3} + 69310 T^{4} - 576225 T^{5} + 69310 p T^{6} - 4911 p^{2} T^{7} + 424 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 10 T + 298 T^{2} - 2663 T^{3} + 41844 T^{4} - 313927 T^{5} + 41844 p T^{6} - 2663 p^{2} T^{7} + 298 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 9 T + 152 T^{2} - 1354 T^{3} + 16110 T^{4} - 77787 T^{5} + 16110 p T^{6} - 1354 p^{2} T^{7} + 152 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 43 T + 1116 T^{2} - 20334 T^{3} + 284864 T^{4} - 3137841 T^{5} + 284864 p T^{6} - 20334 p^{2} T^{7} + 1116 p^{3} T^{8} - 43 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.62408368286137728558312245152, −4.49656824958577474087989028741, −4.47069384254764580548974165020, −4.44334928525471954227239076311, −4.19986198954772765857963111980, −3.83454860044774772107315312131, −3.76733254895306554536255587473, −3.65672359375855363310561009092, −3.46238118379919470880099029264, −3.23347153390004938023677129772, −3.22319149794565069827812470162, −3.02307141003887023466744598125, −2.87882448904508772660662891845, −2.68958200814234590084267798866, −2.33538947723241709668203270038, −2.21827260164598141191331094642, −2.16288368510272806565194972756, −1.99977616940189624227802758974, −1.86503355721047761985493210463, −1.62806701258419936908133302597, −1.08964772200885084770625162334, −1.01541306523134564639355665423, −0.76526092512813862608375468460, −0.75594987391056671760733988397, −0.57261960915105736812628925995, 0.57261960915105736812628925995, 0.75594987391056671760733988397, 0.76526092512813862608375468460, 1.01541306523134564639355665423, 1.08964772200885084770625162334, 1.62806701258419936908133302597, 1.86503355721047761985493210463, 1.99977616940189624227802758974, 2.16288368510272806565194972756, 2.21827260164598141191331094642, 2.33538947723241709668203270038, 2.68958200814234590084267798866, 2.87882448904508772660662891845, 3.02307141003887023466744598125, 3.22319149794565069827812470162, 3.23347153390004938023677129772, 3.46238118379919470880099029264, 3.65672359375855363310561009092, 3.76733254895306554536255587473, 3.83454860044774772107315312131, 4.19986198954772765857963111980, 4.44334928525471954227239076311, 4.47069384254764580548974165020, 4.49656824958577474087989028741, 4.62408368286137728558312245152

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