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Properties

Label 10-9675e5-1.1-c1e5-0-10
Degree $10$
Conductor $8.477\times 10^{19}$
Sign $-1$
Analytic cond. $2.75194\times 10^{9}$
Root an. cond. $8.78949$
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  + 2·2-s + 4-s − 5·7-s − 8-s + 6·11-s − 5·13-s − 10·14-s − 2·16-s − 17·17-s − 6·19-s + 12·22-s + 23-s − 10·26-s − 5·28-s − 6·29-s + 6·31-s − 4·32-s − 34·34-s − 5·37-s − 12·38-s − 2·41-s + 5·43-s + 6·44-s + 2·46-s + 4·49-s − 5·52-s − 23·53-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s − 1.88·7-s − 0.353·8-s + 1.80·11-s − 1.38·13-s − 2.67·14-s − 1/2·16-s − 4.12·17-s − 1.37·19-s + 2.55·22-s + 0.208·23-s − 1.96·26-s − 0.944·28-s − 1.11·29-s + 1.07·31-s − 0.707·32-s − 5.83·34-s − 0.821·37-s − 1.94·38-s − 0.312·41-s + 0.762·43-s + 0.904·44-s + 0.294·46-s + 4/7·49-s − 0.693·52-s − 3.15·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{10} \cdot 43^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{10} \cdot 43^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{10} \cdot 5^{10} \cdot 43^{5}\)
Sign: $-1$
Analytic conductor: \(2.75194\times 10^{9}\)
Root analytic conductor: \(8.78949\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 3^{10} \cdot 5^{10} \cdot 43^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad3 \( 1 \)
5 \( 1 \)
43$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - p T + 3 T^{2} - 3 T^{3} + 3 T^{4} + 3 p T^{6} - 3 p^{2} T^{7} + 3 p^{3} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 5 T + 3 p T^{2} + 43 T^{3} + 138 T^{4} + 272 T^{5} + 138 p T^{6} + 43 p^{2} T^{7} + 3 p^{4} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 6 T + 56 T^{2} - 221 T^{3} + 1184 T^{4} - 3398 T^{5} + 1184 p T^{6} - 221 p^{2} T^{7} + 56 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 5 T + 15 T^{2} - 24 T^{3} - 36 T^{4} - 314 T^{5} - 36 p T^{6} - 24 p^{2} T^{7} + 15 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + p T + 179 T^{2} + 1336 T^{3} + 7764 T^{4} + 35582 T^{5} + 7764 p T^{6} + 1336 p^{2} T^{7} + 179 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 6 T + 23 T^{2} + 104 T^{3} + 786 T^{4} + 4228 T^{5} + 786 p T^{6} + 104 p^{2} T^{7} + 23 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - T + 61 T^{2} + 40 T^{3} + 1764 T^{4} + 2514 T^{5} + 1764 p T^{6} + 40 p^{2} T^{7} + 61 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 6 T + 61 T^{2} - 56 T^{3} - 642 T^{4} - 14492 T^{5} - 642 p T^{6} - 56 p^{2} T^{7} + 61 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 6 T + 88 T^{2} - 215 T^{3} + 2476 T^{4} - 1670 T^{5} + 2476 p T^{6} - 215 p^{2} T^{7} + 88 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 5 T + 157 T^{2} + 613 T^{3} + 10668 T^{4} + 32072 T^{5} + 10668 p T^{6} + 613 p^{2} T^{7} + 157 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 2 T + 106 T^{2} + 81 T^{3} + 4844 T^{4} - 112 T^{5} + 4844 p T^{6} + 81 p^{2} T^{7} + 106 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 111 T^{2} + 72 T^{3} + 7998 T^{4} + 4720 T^{5} + 7998 p T^{6} + 72 p^{2} T^{7} + 111 p^{3} T^{8} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 23 T + 455 T^{2} + 5544 T^{3} + 59212 T^{4} + 458850 T^{5} + 59212 p T^{6} + 5544 p^{2} T^{7} + 455 p^{3} T^{8} + 23 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - T + 279 T^{2} - 229 T^{3} + 32042 T^{4} - 20044 T^{5} + 32042 p T^{6} - 229 p^{2} T^{7} + 279 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 20 T + 225 T^{2} - 1728 T^{3} + 15066 T^{4} - 122648 T^{5} + 15066 p T^{6} - 1728 p^{2} T^{7} + 225 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 21 T + 379 T^{2} + 4896 T^{3} + 54318 T^{4} + 467430 T^{5} + 54318 p T^{6} + 4896 p^{2} T^{7} + 379 p^{3} T^{8} + 21 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 4 T + 143 T^{2} + 504 T^{3} + 15830 T^{4} + 51592 T^{5} + 15830 p T^{6} + 504 p^{2} T^{7} + 143 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 5 T + 281 T^{2} + 1269 T^{3} + 36116 T^{4} + 130872 T^{5} + 36116 p T^{6} + 1269 p^{2} T^{7} + 281 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 41 T + 1039 T^{2} - 17721 T^{3} + 231158 T^{4} - 2306844 T^{5} + 231158 p T^{6} - 17721 p^{2} T^{7} + 1039 p^{3} T^{8} - 41 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 7 T + 317 T^{2} + 1436 T^{3} + 43232 T^{4} + 144330 T^{5} + 43232 p T^{6} + 1436 p^{2} T^{7} + 317 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 20 T + 453 T^{2} + 6120 T^{3} + 80658 T^{4} + 775176 T^{5} + 80658 p T^{6} + 6120 p^{2} T^{7} + 453 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 37 T + 895 T^{2} + 15564 T^{3} + 213240 T^{4} + 2321998 T^{5} + 213240 p T^{6} + 15564 p^{2} T^{7} + 895 p^{3} T^{8} + 37 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.73328328253585954447264761873, −4.52853067894524034996866540462, −4.48459045402210274341811554100, −4.40091532120182130496120216735, −4.38579132887500670812356461529, −4.20727160079272247225689772768, −4.02682295750673391294995696055, −3.83037507679105831350677337218, −3.76239257274226196965436425092, −3.51292085414082513091976791192, −3.37731637866677645856881532597, −3.33575225195025731598704058716, −3.05161930758087049694189732590, −2.86533816001085237761621533084, −2.79839818462736013818345247721, −2.46398144832054772465339081372, −2.46301777148767538527446995086, −2.29731006614570614063272107462, −2.07941578037639182744936319344, −1.82205687459861218436006280995, −1.72892470253258327486028388561, −1.72301527887832867011756768688, −1.11648776391397113141892805255, −1.00069913925632359097518190311, −0.977472810125422684673627037221, 0, 0, 0, 0, 0, 0.977472810125422684673627037221, 1.00069913925632359097518190311, 1.11648776391397113141892805255, 1.72301527887832867011756768688, 1.72892470253258327486028388561, 1.82205687459861218436006280995, 2.07941578037639182744936319344, 2.29731006614570614063272107462, 2.46301777148767538527446995086, 2.46398144832054772465339081372, 2.79839818462736013818345247721, 2.86533816001085237761621533084, 3.05161930758087049694189732590, 3.33575225195025731598704058716, 3.37731637866677645856881532597, 3.51292085414082513091976791192, 3.76239257274226196965436425092, 3.83037507679105831350677337218, 4.02682295750673391294995696055, 4.20727160079272247225689772768, 4.38579132887500670812356461529, 4.40091532120182130496120216735, 4.48459045402210274341811554100, 4.52853067894524034996866540462, 4.73328328253585954447264761873

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