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Properties

Label 10-8016e5-1.1-c1e5-0-5
Degree $10$
Conductor $3.310\times 10^{19}$
Sign $-1$
Analytic cond. $1.07441\times 10^{9}$
Root an. cond. $8.00050$
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  + 5·3-s − 3·5-s + 2·7-s + 15·9-s − 3·11-s − 4·13-s − 15·15-s − 7·17-s + 2·19-s + 10·21-s − 13·23-s − 9·25-s + 35·27-s − 3·29-s + 12·31-s − 15·33-s − 6·35-s − 7·37-s − 20·39-s − 16·41-s − 45·45-s − 47-s − 24·49-s − 35·51-s + 3·53-s + 9·55-s + 10·57-s + ⋯
L(s)  = 1  + 2.88·3-s − 1.34·5-s + 0.755·7-s + 5·9-s − 0.904·11-s − 1.10·13-s − 3.87·15-s − 1.69·17-s + 0.458·19-s + 2.18·21-s − 2.71·23-s − 9/5·25-s + 6.73·27-s − 0.557·29-s + 2.15·31-s − 2.61·33-s − 1.01·35-s − 1.15·37-s − 3.20·39-s − 2.49·41-s − 6.70·45-s − 0.145·47-s − 3.42·49-s − 4.90·51-s + 0.412·53-s + 1.21·55-s + 1.32·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{5} \cdot 167^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{5} \cdot 167^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{20} \cdot 3^{5} \cdot 167^{5}\)
Sign: $-1$
Analytic conductor: \(1.07441\times 10^{9}\)
Root analytic conductor: \(8.00050\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{20} \cdot 3^{5} \cdot 167^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
3$C_1$ \( ( 1 - T )^{5} \)
167$C_1$ \( ( 1 + T )^{5} \)
good5$C_2 \wr S_5$ \( 1 + 3 T + 18 T^{2} + 44 T^{3} + 32 p T^{4} + 293 T^{5} + 32 p^{2} T^{6} + 44 p^{2} T^{7} + 18 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 2 T + 4 p T^{2} - 37 T^{3} + 334 T^{4} - 325 T^{5} + 334 p T^{6} - 37 p^{2} T^{7} + 4 p^{4} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 3 T + 48 T^{2} + 116 T^{3} + 994 T^{4} + 1829 T^{5} + 994 p T^{6} + 116 p^{2} T^{7} + 48 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 4 T + 4 p T^{2} + 10 p T^{3} + 1089 T^{4} + 2017 T^{5} + 1089 p T^{6} + 10 p^{3} T^{7} + 4 p^{4} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 7 T + 63 T^{2} + 258 T^{3} + 1597 T^{4} + 5353 T^{5} + 1597 p T^{6} + 258 p^{2} T^{7} + 63 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 2 T + 54 T^{2} - 66 T^{3} + 1489 T^{4} - 1523 T^{5} + 1489 p T^{6} - 66 p^{2} T^{7} + 54 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 13 T + 128 T^{2} + 964 T^{3} + 6130 T^{4} + 31369 T^{5} + 6130 p T^{6} + 964 p^{2} T^{7} + 128 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 3 T + 24 T^{2} + 38 T^{3} + 958 T^{4} + 4847 T^{5} + 958 p T^{6} + 38 p^{2} T^{7} + 24 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 12 T + 181 T^{2} - 1471 T^{3} + 11998 T^{4} - 68147 T^{5} + 11998 p T^{6} - 1471 p^{2} T^{7} + 181 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 7 T + 50 T^{2} + 162 T^{3} + 2944 T^{4} + 17933 T^{5} + 2944 p T^{6} + 162 p^{2} T^{7} + 50 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 16 T + 181 T^{2} + 967 T^{3} + 4174 T^{4} + 9063 T^{5} + 4174 p T^{6} + 967 p^{2} T^{7} + 181 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 69 T^{2} - 337 T^{3} + 4476 T^{4} - 11541 T^{5} + 4476 p T^{6} - 337 p^{2} T^{7} + 69 p^{3} T^{8} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + T + 110 T^{2} - 139 T^{3} + 4573 T^{4} - 16971 T^{5} + 4573 p T^{6} - 139 p^{2} T^{7} + 110 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 3 T + 192 T^{2} - 751 T^{3} + 16843 T^{4} - 62173 T^{5} + 16843 p T^{6} - 751 p^{2} T^{7} + 192 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + T + 190 T^{2} - p T^{3} + 17725 T^{4} - 10735 T^{5} + 17725 p T^{6} - p^{3} T^{7} + 190 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 22 T + 354 T^{2} + 3558 T^{3} + 32581 T^{4} + 244855 T^{5} + 32581 p T^{6} + 3558 p^{2} T^{7} + 354 p^{3} T^{8} + 22 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 2 T + 210 T^{2} + 50 T^{3} + 20755 T^{4} - 11445 T^{5} + 20755 p T^{6} + 50 p^{2} T^{7} + 210 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 9 T + 249 T^{2} + 1683 T^{3} + 25871 T^{4} + 146855 T^{5} + 25871 p T^{6} + 1683 p^{2} T^{7} + 249 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 28 T + 470 T^{2} + 5474 T^{3} + 55601 T^{4} + 491083 T^{5} + 55601 p T^{6} + 5474 p^{2} T^{7} + 470 p^{3} T^{8} + 28 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 28 T + 535 T^{2} - 7391 T^{3} + 85444 T^{4} - 807623 T^{5} + 85444 p T^{6} - 7391 p^{2} T^{7} + 535 p^{3} T^{8} - 28 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 7 T + 86 T^{2} + 1801 T^{3} + 18723 T^{4} + 86479 T^{5} + 18723 p T^{6} + 1801 p^{2} T^{7} + 86 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 30 T + 642 T^{2} + 9704 T^{3} + 122785 T^{4} + 1246643 T^{5} + 122785 p T^{6} + 9704 p^{2} T^{7} + 642 p^{3} T^{8} + 30 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 33 T + 769 T^{2} + 12555 T^{3} + 168965 T^{4} + 1807507 T^{5} + 168965 p T^{6} + 12555 p^{2} T^{7} + 769 p^{3} T^{8} + 33 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.79243218081290810322459184763, −4.54326553062249836161140627064, −4.51900559763507410253768989833, −4.51233798842809681033009328798, −4.48595467588812545455949874451, −3.94849258508433544508991254954, −3.92986070726547926519775082612, −3.91012237227508719049179502270, −3.87557139421563456363296699906, −3.80454460852184534237914880659, −3.18564734943500164841517593911, −3.18106104095410002540229221697, −3.12351731201298354217245712912, −3.00765604875511026666860698934, −2.81273759436026701072006423451, −2.65699788928315816508872548000, −2.35416264458265570852640161779, −2.30531817413073882752172371719, −2.05303005137794860426699804076, −2.03437510958205305060843040834, −1.68347139007611206627796975741, −1.64281000035156101438619902158, −1.29661037431901013447534413286, −1.25517398044310573664445172492, −1.23974628557578783770049467261, 0, 0, 0, 0, 0, 1.23974628557578783770049467261, 1.25517398044310573664445172492, 1.29661037431901013447534413286, 1.64281000035156101438619902158, 1.68347139007611206627796975741, 2.03437510958205305060843040834, 2.05303005137794860426699804076, 2.30531817413073882752172371719, 2.35416264458265570852640161779, 2.65699788928315816508872548000, 2.81273759436026701072006423451, 3.00765604875511026666860698934, 3.12351731201298354217245712912, 3.18106104095410002540229221697, 3.18564734943500164841517593911, 3.80454460852184534237914880659, 3.87557139421563456363296699906, 3.91012237227508719049179502270, 3.92986070726547926519775082612, 3.94849258508433544508991254954, 4.48595467588812545455949874451, 4.51233798842809681033009328798, 4.51900559763507410253768989833, 4.54326553062249836161140627064, 4.79243218081290810322459184763

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