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Properties

Label 10-2960e5-1.1-c1e5-0-5
Degree $10$
Conductor $2.272\times 10^{17}$
Sign $-1$
Analytic cond. $7.37639\times 10^{6}$
Root an. cond. $4.86165$
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  + 3-s + 5·5-s − 7·7-s − 6·9-s − 7·11-s + 2·13-s + 5·15-s − 8·17-s − 14·19-s − 7·21-s − 2·23-s + 15·25-s − 5·27-s + 2·29-s − 8·31-s − 7·33-s − 35·35-s − 5·37-s + 2·39-s − 9·41-s − 14·43-s − 30·45-s + 5·47-s + 8·49-s − 8·51-s − 15·53-s − 35·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 2.23·5-s − 2.64·7-s − 2·9-s − 2.11·11-s + 0.554·13-s + 1.29·15-s − 1.94·17-s − 3.21·19-s − 1.52·21-s − 0.417·23-s + 3·25-s − 0.962·27-s + 0.371·29-s − 1.43·31-s − 1.21·33-s − 5.91·35-s − 0.821·37-s + 0.320·39-s − 1.40·41-s − 2.13·43-s − 4.47·45-s + 0.729·47-s + 8/7·49-s − 1.12·51-s − 2.06·53-s − 4.71·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{5} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{5} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{20} \cdot 5^{5} \cdot 37^{5}\)
Sign: $-1$
Analytic conductor: \(7.37639\times 10^{6}\)
Root analytic conductor: \(4.86165\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{20} \cdot 5^{5} \cdot 37^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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$C_1$ \( ( 1 - T )^{5} \)
37$C_1$ \( ( 1 + T )^{5} \)
good3$C_2 \wr S_5$ \( 1 - T + 7 T^{2} - 8 T^{3} + 22 T^{4} - 32 T^{5} + 22 p T^{6} - 8 p^{2} T^{7} + 7 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + p T + 41 T^{2} + 172 T^{3} + 88 p T^{4} + 1724 T^{5} + 88 p^{2} T^{6} + 172 p^{2} T^{7} + 41 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 7 T + 43 T^{2} + 164 T^{3} + 58 p T^{4} + 1946 T^{5} + 58 p^{2} T^{6} + 164 p^{2} T^{7} + 43 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 2 T + 45 T^{2} - 84 T^{3} + 986 T^{4} - 1596 T^{5} + 986 p T^{6} - 84 p^{2} T^{7} + 45 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 8 T + 97 T^{2} + 508 T^{3} + 3410 T^{4} + 12616 T^{5} + 3410 p T^{6} + 508 p^{2} T^{7} + 97 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 14 T + 121 T^{2} + 702 T^{3} + 3310 T^{4} + 14344 T^{5} + 3310 p T^{6} + 702 p^{2} T^{7} + 121 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 2 T + 59 T^{2} + 8 p T^{3} + 2306 T^{4} + 4844 T^{5} + 2306 p T^{6} + 8 p^{3} T^{7} + 59 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 2 T + 65 T^{2} + 40 T^{3} + 1274 T^{4} + 5716 T^{5} + 1274 p T^{6} + 40 p^{2} T^{7} + 65 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 8 T + 117 T^{2} + 678 T^{3} + 6422 T^{4} + 29676 T^{5} + 6422 p T^{6} + 678 p^{2} T^{7} + 117 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 9 T + 141 T^{2} + 1172 T^{3} + 10426 T^{4} + 64918 T^{5} + 10426 p T^{6} + 1172 p^{2} T^{7} + 141 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 14 T + 199 T^{2} + 1848 T^{3} + 15994 T^{4} + 108468 T^{5} + 15994 p T^{6} + 1848 p^{2} T^{7} + 199 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 5 T + 121 T^{2} - 1068 T^{3} + 7756 T^{4} - 74308 T^{5} + 7756 p T^{6} - 1068 p^{2} T^{7} + 121 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 15 T + 329 T^{2} + 3220 T^{3} + 38154 T^{4} + 257034 T^{5} + 38154 p T^{6} + 3220 p^{2} T^{7} + 329 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 12 T + 201 T^{2} + 2306 T^{3} + 22014 T^{4} + 183108 T^{5} + 22014 p T^{6} + 2306 p^{2} T^{7} + 201 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 12 T + 273 T^{2} - 2752 T^{3} + 31594 T^{4} - 246952 T^{5} + 31594 p T^{6} - 2752 p^{2} T^{7} + 273 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 2 T + 161 T^{2} - 466 T^{3} + 17082 T^{4} - 36920 T^{5} + 17082 p T^{6} - 466 p^{2} T^{7} + 161 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 13 T + 7 T^{2} - 484 T^{3} + 4638 T^{4} + 91342 T^{5} + 4638 p T^{6} - 484 p^{2} T^{7} + 7 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 5 T + 297 T^{2} + 1124 T^{3} + 38414 T^{4} + 110990 T^{5} + 38414 p T^{6} + 1124 p^{2} T^{7} + 297 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 36 T + 825 T^{2} + 13314 T^{3} + 165914 T^{4} + 1648348 T^{5} + 165914 p T^{6} + 13314 p^{2} T^{7} + 825 p^{3} T^{8} + 36 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 9 T + 311 T^{2} - 1948 T^{3} + 41166 T^{4} - 199680 T^{5} + 41166 p T^{6} - 1948 p^{2} T^{7} + 311 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 16 T + 205 T^{2} + 560 T^{3} - 5878 T^{4} - 151936 T^{5} - 5878 p T^{6} + 560 p^{2} T^{7} + 205 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 21 T^{2} + 496 T^{3} + 3450 T^{4} - 97184 T^{5} + 3450 p T^{6} + 496 p^{2} T^{7} + 21 p^{3} T^{8} + p^{5} T^{10} \)
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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

−5.58028239301503474909645792831, −5.35618980681432427587515435272, −5.35338465242232259846488662346, −5.22846951701874374382315373650, −5.09665827539631837213274626325, −4.85686172483106843524723320801, −4.68761941723794806605671379960, −4.38001477391335090909706230105, −4.37468426671051995729586432162, −4.09413750010268114460292356372, −3.76737891892066858123119680794, −3.52263289477318909486113188644, −3.48255911790158320841692591742, −3.40150999301035086235205884531, −3.13467021174521489653211773253, −2.79332770799495244924705961614, −2.75929513531011371824519780020, −2.50692114123233260066549481481, −2.45848808150595997945896068719, −2.44304664785494896827526769652, −2.18701206860746762720220146214, −1.74521948888927097265488773365, −1.65350863086542394974810983568, −1.37523077511785640567324497004, −1.27601667716178738106085765511, 0, 0, 0, 0, 0, 1.27601667716178738106085765511, 1.37523077511785640567324497004, 1.65350863086542394974810983568, 1.74521948888927097265488773365, 2.18701206860746762720220146214, 2.44304664785494896827526769652, 2.45848808150595997945896068719, 2.50692114123233260066549481481, 2.75929513531011371824519780020, 2.79332770799495244924705961614, 3.13467021174521489653211773253, 3.40150999301035086235205884531, 3.48255911790158320841692591742, 3.52263289477318909486113188644, 3.76737891892066858123119680794, 4.09413750010268114460292356372, 4.37468426671051995729586432162, 4.38001477391335090909706230105, 4.68761941723794806605671379960, 4.85686172483106843524723320801, 5.09665827539631837213274626325, 5.22846951701874374382315373650, 5.35338465242232259846488662346, 5.35618980681432427587515435272, 5.58028239301503474909645792831

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