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Properties

Label 10-8036e5-1.1-c1e5-0-2
Degree $10$
Conductor $3.351\times 10^{19}$
Sign $-1$
Analytic cond. $1.08789\times 10^{9}$
Root an. cond. $8.01047$
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  + 2·3-s + 5-s − 5·9-s − 2·11-s + 13-s + 2·15-s + 3·17-s + 4·19-s − 8·23-s − 15·25-s − 12·27-s − 9·29-s + 11·31-s − 4·33-s − 11·37-s + 2·39-s − 5·41-s − 27·43-s − 5·45-s + 3·47-s + 6·51-s − 19·53-s − 2·55-s + 8·57-s + 15·59-s + 65-s − 21·67-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s − 5/3·9-s − 0.603·11-s + 0.277·13-s + 0.516·15-s + 0.727·17-s + 0.917·19-s − 1.66·23-s − 3·25-s − 2.30·27-s − 1.67·29-s + 1.97·31-s − 0.696·33-s − 1.80·37-s + 0.320·39-s − 0.780·41-s − 4.11·43-s − 0.745·45-s + 0.437·47-s + 0.840·51-s − 2.60·53-s − 0.269·55-s + 1.05·57-s + 1.95·59-s + 0.124·65-s − 2.56·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 7^{10} \cdot 41^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 7^{10} \cdot 41^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{10} \cdot 7^{10} \cdot 41^{5}\)
Sign: $-1$
Analytic conductor: \(1.08789\times 10^{9}\)
Root analytic conductor: \(8.01047\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{10} \cdot 7^{10} \cdot 41^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
7 \( 1 \)
41$C_1$ \( ( 1 + T )^{5} \)
good3$C_2 \wr S_5$ \( 1 - 2 T + p^{2} T^{2} - 16 T^{3} + 43 T^{4} - 59 T^{5} + 43 p T^{6} - 16 p^{2} T^{7} + p^{5} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 - T + 16 T^{2} - 12 T^{3} + 127 T^{4} - 78 T^{5} + 127 p T^{6} - 12 p^{2} T^{7} + 16 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 2 T + 26 T^{2} - 10 T^{3} + 15 p T^{4} - 728 T^{5} + 15 p^{2} T^{6} - 10 p^{2} T^{7} + 26 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - T + 42 T^{2} - 4 T^{3} + 835 T^{4} + 131 T^{5} + 835 p T^{6} - 4 p^{2} T^{7} + 42 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 3 T + 56 T^{2} - 126 T^{3} + 1591 T^{4} - 3013 T^{5} + 1591 p T^{6} - 126 p^{2} T^{7} + 56 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 4 T + 39 T^{2} - 48 T^{3} + 441 T^{4} + 365 T^{5} + 441 p T^{6} - 48 p^{2} T^{7} + 39 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 8 T + 83 T^{2} + 574 T^{3} + 3355 T^{4} + 18011 T^{5} + 3355 p T^{6} + 574 p^{2} T^{7} + 83 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 9 T + 68 T^{2} + 24 T^{3} - 1301 T^{4} - 16106 T^{5} - 1301 p T^{6} + 24 p^{2} T^{7} + 68 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 11 T + 112 T^{2} - 656 T^{3} + 5003 T^{4} - 25610 T^{5} + 5003 p T^{6} - 656 p^{2} T^{7} + 112 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 11 T + 145 T^{2} + 663 T^{3} + 4956 T^{4} + 12852 T^{5} + 4956 p T^{6} + 663 p^{2} T^{7} + 145 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 27 T + 382 T^{2} + 3986 T^{3} + 34437 T^{4} + 248301 T^{5} + 34437 p T^{6} + 3986 p^{2} T^{7} + 382 p^{3} T^{8} + 27 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 3 T + 167 T^{2} - 461 T^{3} + 13280 T^{4} - 29348 T^{5} + 13280 p T^{6} - 461 p^{2} T^{7} + 167 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 19 T + 298 T^{2} + 2774 T^{3} + 25329 T^{4} + 175326 T^{5} + 25329 p T^{6} + 2774 p^{2} T^{7} + 298 p^{3} T^{8} + 19 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 15 T + 316 T^{2} - 3050 T^{3} + 36379 T^{4} - 253030 T^{5} + 36379 p T^{6} - 3050 p^{2} T^{7} + 316 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 190 T^{2} + 30 T^{3} + 18957 T^{4} + 5364 T^{5} + 18957 p T^{6} + 30 p^{2} T^{7} + 190 p^{3} T^{8} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 21 T + 310 T^{2} + 3174 T^{3} + 31197 T^{4} + 258962 T^{5} + 31197 p T^{6} + 3174 p^{2} T^{7} + 310 p^{3} T^{8} + 21 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 16 T + 338 T^{2} + 4312 T^{3} + 46713 T^{4} + 451448 T^{5} + 46713 p T^{6} + 4312 p^{2} T^{7} + 338 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 10 T + 222 T^{2} - 2492 T^{3} + 25941 T^{4} - 257572 T^{5} + 25941 p T^{6} - 2492 p^{2} T^{7} + 222 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 14 T + 315 T^{2} + 3176 T^{3} + 45082 T^{4} + 346676 T^{5} + 45082 p T^{6} + 3176 p^{2} T^{7} + 315 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 2 T + 114 T^{2} - 338 T^{3} + 12089 T^{4} - 78912 T^{5} + 12089 p T^{6} - 338 p^{2} T^{7} + 114 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 6 T + 305 T^{2} + 1284 T^{3} + 45117 T^{4} + 153431 T^{5} + 45117 p T^{6} + 1284 p^{2} T^{7} + 305 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 20 T + 207 T^{2} + 2236 T^{3} + 33747 T^{4} + 423407 T^{5} + 33747 p T^{6} + 2236 p^{2} T^{7} + 207 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.11052390457124230291978306647, −5.02223808619525378764179743943, −4.65545845687515627479167429788, −4.58557452562690562567536837168, −4.17884562520955826651471673875, −4.02084767271942798378072017926, −3.96002341893884566105047271203, −3.87893233366490782683902218274, −3.79500091288163269975730009114, −3.67234208480226290576368862286, −3.19900247903349388000085560851, −3.10210383309254710870534536813, −3.05235699423200649684671778234, −3.03424719297058183123810829750, −3.00860999259595689937906605899, −2.51841585944807750910664689056, −2.47295617542385002919829168519, −2.27998151601237220621441626752, −2.01973904219962935501905225809, −1.98069127690133780787690493883, −1.74349039067530723182622230281, −1.45652610264354766190205313993, −1.31711524114578615084173497476, −1.27405881891230418732113040237, −1.03009251481172247951656997382, 0, 0, 0, 0, 0, 1.03009251481172247951656997382, 1.27405881891230418732113040237, 1.31711524114578615084173497476, 1.45652610264354766190205313993, 1.74349039067530723182622230281, 1.98069127690133780787690493883, 2.01973904219962935501905225809, 2.27998151601237220621441626752, 2.47295617542385002919829168519, 2.51841585944807750910664689056, 3.00860999259595689937906605899, 3.03424719297058183123810829750, 3.05235699423200649684671778234, 3.10210383309254710870534536813, 3.19900247903349388000085560851, 3.67234208480226290576368862286, 3.79500091288163269975730009114, 3.87893233366490782683902218274, 3.96002341893884566105047271203, 4.02084767271942798378072017926, 4.17884562520955826651471673875, 4.58557452562690562567536837168, 4.65545845687515627479167429788, 5.02223808619525378764179743943, 5.11052390457124230291978306647

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