Properties

Label 10-6012e5-1.1-c1e5-0-2
Degree $10$
Conductor $7.854\times 10^{18}$
Sign $-1$
Analytic cond. $2.54964\times 10^{8}$
Root an. cond. $6.92864$
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  + 3·5-s − 2·7-s − 3·11-s − 4·13-s + 7·17-s − 2·19-s − 13·23-s − 9·25-s + 3·29-s − 12·31-s − 6·35-s − 7·37-s + 16·41-s − 47-s − 24·49-s − 3·53-s − 9·55-s − 59-s − 22·61-s − 12·65-s + 2·67-s − 9·71-s − 28·73-s + 6·77-s − 28·79-s − 7·83-s + 21·85-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.755·7-s − 0.904·11-s − 1.10·13-s + 1.69·17-s − 0.458·19-s − 2.71·23-s − 9/5·25-s + 0.557·29-s − 2.15·31-s − 1.01·35-s − 1.15·37-s + 2.49·41-s − 0.145·47-s − 3.42·49-s − 0.412·53-s − 1.21·55-s − 0.130·59-s − 2.81·61-s − 1.48·65-s + 0.244·67-s − 1.06·71-s − 3.27·73-s + 0.683·77-s − 3.15·79-s − 0.768·83-s + 2.27·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 167^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 167^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{10} \cdot 3^{10} \cdot 167^{5}\)
Sign: $-1$
Analytic conductor: \(2.54964\times 10^{8}\)
Root analytic conductor: \(6.92864\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{10} \cdot 3^{10} \cdot 167^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
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

−5.16955649415537615426162004414, −4.88616510952164231173960217006, −4.84867719900937316276912571721, −4.80721138631324022318402012982, −4.62686996054966803506194978026, −4.19974539246401247614569251246, −4.13792123369276741832255580400, −4.06808840800925903298221248632, −3.99389401966633799739567081359, −3.82992566428029633323551609650, −3.36612142901953290951722349151, −3.27989740373928837194488072081, −3.23885387181180400624779964981, −3.20531920475735990823439026240, −2.87017305111885516520118325037, −2.53238077682676441617042125081, −2.45739273861161578006893434248, −2.37004923225786333115017114363, −2.16858762481475496912894974238, −2.12732515059847328073644633194, −1.60127660390587798453135279768, −1.50289929976183186546055254327, −1.40926218852153424849170099262, −1.23292041884592317070890126957, −1.23290122109438951178039872158, 0, 0, 0, 0, 0, 1.23290122109438951178039872158, 1.23292041884592317070890126957, 1.40926218852153424849170099262, 1.50289929976183186546055254327, 1.60127660390587798453135279768, 2.12732515059847328073644633194, 2.16858762481475496912894974238, 2.37004923225786333115017114363, 2.45739273861161578006893434248, 2.53238077682676441617042125081, 2.87017305111885516520118325037, 3.20531920475735990823439026240, 3.23885387181180400624779964981, 3.27989740373928837194488072081, 3.36612142901953290951722349151, 3.82992566428029633323551609650, 3.99389401966633799739567081359, 4.06808840800925903298221248632, 4.13792123369276741832255580400, 4.19974539246401247614569251246, 4.62686996054966803506194978026, 4.80721138631324022318402012982, 4.84867719900937316276912571721, 4.88616510952164231173960217006, 5.16955649415537615426162004414

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