↑

Properties

Label 10-63e10-1.1-c1e5-0-3
Degree $10$
Conductor $9.849\times 10^{17}$
Sign $-1$
Analytic cond. $3.19735\times 10^{7}$
Root an. cond. $5.62962$
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  + 2·2-s − 4-s − 4·5-s − 5·8-s − 8·10-s + 4·11-s − 8·13-s − 4·16-s − 12·17-s + 19-s + 4·20-s + 8·22-s + 3·23-s − 4·25-s − 16·26-s + 7·29-s − 3·31-s − 32-s − 24·34-s + 2·38-s + 20·40-s − 5·41-s + 7·43-s − 4·44-s + 6·46-s − 27·47-s − 8·50-s + ⋯
L(s)  = 1  + 1.41·2-s − 1/2·4-s − 1.78·5-s − 1.76·8-s − 2.52·10-s + 1.20·11-s − 2.21·13-s − 16-s − 2.91·17-s + 0.229·19-s + 0.894·20-s + 1.70·22-s + 0.625·23-s − 4/5·25-s − 3.13·26-s + 1.29·29-s − 0.538·31-s − 0.176·32-s − 4.11·34-s + 0.324·38-s + 3.16·40-s − 0.780·41-s + 1.06·43-s − 0.603·44-s + 0.884·46-s − 3.93·47-s − 1.13·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{20} \cdot 7^{10}\)
Sign: $-1$
Analytic conductor: \(3.19735\times 10^{7}\)
Root analytic conductor: \(5.62962\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 3^{20} \cdot 7^{10} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
7 \( 1 \)
good2$C_2 \wr S_5$ \( 1 - p T + 5 T^{2} - 7 T^{3} + 13 T^{4} - 15 T^{5} + 13 p T^{6} - 7 p^{2} T^{7} + 5 p^{3} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 4 T + 4 p T^{2} + 62 T^{3} + 193 T^{4} + 423 T^{5} + 193 p T^{6} + 62 p^{2} T^{7} + 4 p^{4} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 4 T + 47 T^{2} - 161 T^{3} + 958 T^{4} - 2589 T^{5} + 958 p T^{6} - 161 p^{2} T^{7} + 47 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 8 T + 6 p T^{2} + 31 p T^{3} + 2174 T^{4} + 7779 T^{5} + 2174 p T^{6} + 31 p^{3} T^{7} + 6 p^{4} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 12 T + 130 T^{2} + 876 T^{3} + 5203 T^{4} + 22839 T^{5} + 5203 p T^{6} + 876 p^{2} T^{7} + 130 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - T + 54 T^{2} - 122 T^{3} + 1532 T^{4} - 3483 T^{5} + 1532 p T^{6} - 122 p^{2} T^{7} + 54 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 3 T + 52 T^{2} - 225 T^{3} + 2023 T^{4} - 5565 T^{5} + 2023 p T^{6} - 225 p^{2} T^{7} + 52 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 7 T + 125 T^{2} - 647 T^{3} + 6535 T^{4} - 25761 T^{5} + 6535 p T^{6} - 647 p^{2} T^{7} + 125 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 3 T + 134 T^{2} + 308 T^{3} + 250 p T^{4} + 13615 T^{5} + 250 p^{2} T^{6} + 308 p^{2} T^{7} + 134 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 89 T^{2} + 280 T^{3} + 3418 T^{4} + 20432 T^{5} + 3418 p T^{6} + 280 p^{2} T^{7} + 89 p^{3} T^{8} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 5 T + 161 T^{2} + 769 T^{3} + 11569 T^{4} + 46293 T^{5} + 11569 p T^{6} + 769 p^{2} T^{7} + 161 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 7 T + 126 T^{2} - 474 T^{3} + 5544 T^{4} - 14049 T^{5} + 5544 p T^{6} - 474 p^{2} T^{7} + 126 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 27 T + 448 T^{2} + 5169 T^{3} + 48091 T^{4} + 359985 T^{5} + 48091 p T^{6} + 5169 p^{2} T^{7} + 448 p^{3} T^{8} + 27 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 21 T + 400 T^{2} + 4662 T^{3} + 49132 T^{4} + 375771 T^{5} + 49132 p T^{6} + 4662 p^{2} T^{7} + 400 p^{3} T^{8} + 21 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 30 T + 601 T^{2} + 8193 T^{3} + 88864 T^{4} + 752289 T^{5} + 88864 p T^{6} + 8193 p^{2} T^{7} + 601 p^{3} T^{8} + 30 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 14 T + 339 T^{2} + 3409 T^{3} + 43418 T^{4} + 311709 T^{5} + 43418 p T^{6} + 3409 p^{2} T^{7} + 339 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 2 T + 132 T^{2} - 196 T^{3} + 10871 T^{4} - 15429 T^{5} + 10871 p T^{6} - 196 p^{2} T^{7} + 132 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 3 T + 187 T^{2} - 285 T^{3} + 15679 T^{4} - 10143 T^{5} + 15679 p T^{6} - 285 p^{2} T^{7} + 187 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 15 T + 359 T^{2} - 3943 T^{3} + 53173 T^{4} - 414929 T^{5} + 53173 p T^{6} - 3943 p^{2} T^{7} + 359 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 4 T + 300 T^{2} - 1488 T^{3} + 39873 T^{4} - 184983 T^{5} + 39873 p T^{6} - 1488 p^{2} T^{7} + 300 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 9 T + 229 T^{2} + 882 T^{3} + 19849 T^{4} + 37179 T^{5} + 19849 p T^{6} + 882 p^{2} T^{7} + 229 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 28 T + 680 T^{2} + 10388 T^{3} + 140263 T^{4} + 1402827 T^{5} + 140263 p T^{6} + 10388 p^{2} T^{7} + 680 p^{3} T^{8} + 28 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 12 T + 341 T^{2} + 29 p T^{3} + 54712 T^{4} + 367945 T^{5} + 54712 p T^{6} + 29 p^{3} T^{7} + 341 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
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

−5.10903557151675273994603797756, −4.95134767220609851032191508371, −4.94944290634797755331053225530, −4.85054597505515003515145859566, −4.80807615863779403643212652245, −4.58999008008049951969389806943, −4.33470203604797092807782067362, −4.32676518441506363380620834010, −4.24573745753113610024927233132, −4.19979983518216691161109426105, −3.73581236356338194489505751567, −3.61916842854129136330697708327, −3.56425236403471406447648887430, −3.40026207134217458061118638171, −3.11345435797510662491651847263, −2.90054935844474876370794073679, −2.87232531793158715631217213207, −2.70881537030286740238021555336, −2.19610854423331482959055492894, −2.19188048305630794255347557399, −1.87778708577922697362083449076, −1.86555307451704709761509825743, −1.38465464911110014337151867614, −1.16716644629412749569384251693, −1.10827432486791077545990692997, 0, 0, 0, 0, 0, 1.10827432486791077545990692997, 1.16716644629412749569384251693, 1.38465464911110014337151867614, 1.86555307451704709761509825743, 1.87778708577922697362083449076, 2.19188048305630794255347557399, 2.19610854423331482959055492894, 2.70881537030286740238021555336, 2.87232531793158715631217213207, 2.90054935844474876370794073679, 3.11345435797510662491651847263, 3.40026207134217458061118638171, 3.56425236403471406447648887430, 3.61916842854129136330697708327, 3.73581236356338194489505751567, 4.19979983518216691161109426105, 4.24573745753113610024927233132, 4.32676518441506363380620834010, 4.33470203604797092807782067362, 4.58999008008049951969389806943, 4.80807615863779403643212652245, 4.85054597505515003515145859566, 4.94944290634797755331053225530, 4.95134767220609851032191508371, 5.10903557151675273994603797756

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