↑

Properties

Label 10-8016e5-1.1-c1e5-0-3
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5·3-s − 5-s − 9·7-s + 15·9-s + 2·11-s + 6·13-s + 5·15-s + 4·17-s − 12·19-s + 45·21-s − 2·23-s − 5·25-s − 35·27-s − 6·29-s − 9·31-s − 10·33-s + 9·35-s + 5·37-s − 30·39-s + 4·41-s − 18·43-s − 15·45-s + 7·47-s + 31·49-s − 20·51-s + 3·53-s − 2·55-s + ⋯
L(s)  = 1  − 2.88·3-s − 0.447·5-s − 3.40·7-s + 5·9-s + 0.603·11-s + 1.66·13-s + 1.29·15-s + 0.970·17-s − 2.75·19-s + 9.81·21-s − 0.417·23-s − 25-s − 6.73·27-s − 1.11·29-s − 1.61·31-s − 1.74·33-s + 1.52·35-s + 0.821·37-s − 4.80·39-s + 0.624·41-s − 2.74·43-s − 2.23·45-s + 1.02·47-s + 31/7·49-s − 2.80·51-s + 0.412·53-s − 0.269·55-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$C_2 \wr S_5$ \( 1 + T + 6 T^{2} - T^{3} + p^{2} T^{4} - 36 T^{5} + p^{3} T^{6} - p^{2} T^{7} + 6 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 9 T + 50 T^{2} + 29 p T^{3} + 673 T^{4} + 1888 T^{5} + 673 p T^{6} + 29 p^{3} T^{7} + 50 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 2 T + 27 T^{2} - 28 T^{3} + 430 T^{4} - 420 T^{5} + 430 p T^{6} - 28 p^{2} T^{7} + 27 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 6 T + 63 T^{2} - 20 p T^{3} + 1564 T^{4} - 4740 T^{5} + 1564 p T^{6} - 20 p^{3} T^{7} + 63 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 4 T + 75 T^{2} - 248 T^{3} + 2404 T^{4} - 6144 T^{5} + 2404 p T^{6} - 248 p^{2} T^{7} + 75 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 12 T + 123 T^{2} + 812 T^{3} + 4918 T^{4} + 22368 T^{5} + 4918 p T^{6} + 812 p^{2} T^{7} + 123 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 2 T + 87 T^{2} + 124 T^{3} + 3502 T^{4} + 3876 T^{5} + 3502 p T^{6} + 124 p^{2} T^{7} + 87 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 6 T + 61 T^{2} + 396 T^{3} + 2830 T^{4} + 11148 T^{5} + 2830 p T^{6} + 396 p^{2} T^{7} + 61 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 9 T + 150 T^{2} + 887 T^{3} + 8569 T^{4} + 37440 T^{5} + 8569 p T^{6} + 887 p^{2} T^{7} + 150 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 5 T + 118 T^{2} - 325 T^{3} + 5909 T^{4} - 10586 T^{5} + 5909 p T^{6} - 325 p^{2} T^{7} + 118 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 4 T + 95 T^{2} - 788 T^{3} + 5080 T^{4} - 46992 T^{5} + 5080 p T^{6} - 788 p^{2} T^{7} + 95 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 18 T + 191 T^{2} + 1052 T^{3} + 3586 T^{4} + 6484 T^{5} + 3586 p T^{6} + 1052 p^{2} T^{7} + 191 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 7 T + 174 T^{2} - 1073 T^{3} + 14593 T^{4} - 69780 T^{5} + 14593 p T^{6} - 1073 p^{2} T^{7} + 174 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 3 T + 56 T^{2} + 81 T^{3} + 5467 T^{4} - 17952 T^{5} + 5467 p T^{6} + 81 p^{2} T^{7} + 56 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 13 T + 168 T^{2} + 773 T^{3} + 5239 T^{4} + 4014 T^{5} + 5239 p T^{6} + 773 p^{2} T^{7} + 168 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 16 T + 277 T^{2} - 2880 T^{3} + 32214 T^{4} - 247136 T^{5} + 32214 p T^{6} - 2880 p^{2} T^{7} + 277 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 9 T + 180 T^{2} + 827 T^{3} + 16051 T^{4} + 66492 T^{5} + 16051 p T^{6} + 827 p^{2} T^{7} + 180 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 10 T + 171 T^{2} - 1364 T^{3} + 15538 T^{4} - 96324 T^{5} + 15538 p T^{6} - 1364 p^{2} T^{7} + 171 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 8 T + 261 T^{2} - 1684 T^{3} + 32642 T^{4} - 163368 T^{5} + 32642 p T^{6} - 1684 p^{2} T^{7} + 261 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 6 T + 293 T^{2} + 1088 T^{3} + 38140 T^{4} + 103348 T^{5} + 38140 p T^{6} + 1088 p^{2} T^{7} + 293 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + T + 134 T^{2} + 167 T^{3} + 15877 T^{4} + 3678 T^{5} + 15877 p T^{6} + 167 p^{2} T^{7} + 134 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 5 T + 240 T^{2} + 643 T^{3} + 33427 T^{4} + 85512 T^{5} + 33427 p T^{6} + 643 p^{2} T^{7} + 240 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 7 T + 232 T^{2} + 881 T^{3} + 26207 T^{4} + 67036 T^{5} + 26207 p T^{6} + 881 p^{2} T^{7} + 232 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.05128842683093640256720225432, −4.89754354572355608462360623172, −4.64407472137369257465940749424, −4.53637082553685938691994057766, −4.25204764004940416629834201147, −4.12895164254322328237576500103, −4.02822315538977082946708610615, −3.89481545363778758319792700499, −3.87593061933093898498536378537, −3.78843120829777776433268160995, −3.42287636059051352020182597715, −3.25114028696563933442572696232, −3.23956195792552551733624664473, −3.11306508177960060017045051779, −3.03491549210953838937818667781, −2.46607946211698665377429602429, −2.25306475985968951651942123574, −2.12818723407021074278669266929, −2.05296135066200825623659569026, −2.00789042478887717215826776310, −1.46653185287160632289353459408, −1.26954198384440586475909508206, −1.11101056466128181940294331970, −1.03386322367787766371706180605, −0.78566999108452655594840526518, 0, 0, 0, 0, 0, 0.78566999108452655594840526518, 1.03386322367787766371706180605, 1.11101056466128181940294331970, 1.26954198384440586475909508206, 1.46653185287160632289353459408, 2.00789042478887717215826776310, 2.05296135066200825623659569026, 2.12818723407021074278669266929, 2.25306475985968951651942123574, 2.46607946211698665377429602429, 3.03491549210953838937818667781, 3.11306508177960060017045051779, 3.23956195792552551733624664473, 3.25114028696563933442572696232, 3.42287636059051352020182597715, 3.78843120829777776433268160995, 3.87593061933093898498536378537, 3.89481545363778758319792700499, 4.02822315538977082946708610615, 4.12895164254322328237576500103, 4.25204764004940416629834201147, 4.53637082553685938691994057766, 4.64407472137369257465940749424, 4.89754354572355608462360623172, 5.05128842683093640256720225432

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