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Properties

Label 10-804e5-1.1-c1e5-0-0
Degree $10$
Conductor $3.360\times 10^{14}$
Sign $1$
Analytic cond. $10906.0$
Root an. cond. $2.53376$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·3-s + 3·5-s + 5·7-s + 15·9-s + 6·11-s + 4·13-s + 15·15-s + 17-s + 13·19-s + 25·21-s − 3·25-s + 35·27-s + 3·29-s + 3·31-s + 30·33-s + 15·35-s + 12·37-s + 20·39-s + 11·41-s + 3·43-s + 45·45-s − 13·47-s + 7·49-s + 5·51-s − 9·53-s + 18·55-s + 65·57-s + ⋯
L(s)  = 1  + 2.88·3-s + 1.34·5-s + 1.88·7-s + 5·9-s + 1.80·11-s + 1.10·13-s + 3.87·15-s + 0.242·17-s + 2.98·19-s + 5.45·21-s − 3/5·25-s + 6.73·27-s + 0.557·29-s + 0.538·31-s + 5.22·33-s + 2.53·35-s + 1.97·37-s + 3.20·39-s + 1.71·41-s + 0.457·43-s + 6.70·45-s − 1.89·47-s + 49-s + 0.700·51-s − 1.23·53-s + 2.42·55-s + 8.60·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{5} \cdot 67^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{5} \cdot 67^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{10} \cdot 3^{5} \cdot 67^{5}\)
Sign: $1$
Analytic conductor: \(10906.0\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{10} \cdot 3^{5} \cdot 67^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(42.94582851\)
\(L(\frac12)\) \(\approx\) \(42.94582851\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{5} \)
67$C_1$ \( ( 1 - T )^{5} \)
good5$C_2 \wr S_5$ \( 1 - 3 T + 12 T^{2} - 23 T^{3} + 91 T^{4} - 176 T^{5} + 91 p T^{6} - 23 p^{2} T^{7} + 12 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 5 T + 18 T^{2} - 51 T^{3} + 19 p T^{4} - 288 T^{5} + 19 p^{2} T^{6} - 51 p^{2} T^{7} + 18 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 6 T + 39 T^{2} - 164 T^{3} + 754 T^{4} - 2540 T^{5} + 754 p T^{6} - 164 p^{2} T^{7} + 39 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 4 T + 41 T^{2} - 148 T^{3} + 922 T^{4} - 2464 T^{5} + 922 p T^{6} - 148 p^{2} T^{7} + 41 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - T + 21 T^{2} - 24 T^{3} + 478 T^{4} - 1366 T^{5} + 478 p T^{6} - 24 p^{2} T^{7} + 21 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 13 T + 99 T^{2} - 572 T^{3} + 3194 T^{4} - 15246 T^{5} + 3194 p T^{6} - 572 p^{2} T^{7} + 99 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 55 T^{2} + 72 T^{3} + 1447 T^{4} + 3474 T^{5} + 1447 p T^{6} + 72 p^{2} T^{7} + 55 p^{3} T^{8} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 3 T + 3 p T^{2} - 212 T^{3} + 4036 T^{4} - 8402 T^{5} + 4036 p T^{6} - 212 p^{2} T^{7} + 3 p^{4} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 3 T + 18 T^{2} + 11 T^{3} + 705 T^{4} - 4656 T^{5} + 705 p T^{6} + 11 p^{2} T^{7} + 18 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 12 T + 193 T^{2} - 1530 T^{3} + 13997 T^{4} - 80018 T^{5} + 13997 p T^{6} - 1530 p^{2} T^{7} + 193 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 11 T + 94 T^{2} - 341 T^{3} + 2905 T^{4} - 13348 T^{5} + 2905 p T^{6} - 341 p^{2} T^{7} + 94 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 3 T + 108 T^{2} - 133 T^{3} + 5901 T^{4} - 1440 T^{5} + 5901 p T^{6} - 133 p^{2} T^{7} + 108 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 13 T + 123 T^{2} + 600 T^{3} + 3814 T^{4} + 18094 T^{5} + 3814 p T^{6} + 600 p^{2} T^{7} + 123 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 9 T + 240 T^{2} + 1675 T^{3} + 24397 T^{4} + 126916 T^{5} + 24397 p T^{6} + 1675 p^{2} T^{7} + 240 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 12 T + 253 T^{2} + 2696 T^{3} + 27307 T^{4} + 234602 T^{5} + 27307 p T^{6} + 2696 p^{2} T^{7} + 253 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 4 T + 137 T^{2} - 628 T^{3} + 12754 T^{4} - 55984 T^{5} + 12754 p T^{6} - 628 p^{2} T^{7} + 137 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 24 T + 555 T^{2} + 7492 T^{3} + 93850 T^{4} + 822088 T^{5} + 93850 p T^{6} + 7492 p^{2} T^{7} + 555 p^{3} T^{8} + 24 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 12 T + 357 T^{2} - 3118 T^{3} + 50553 T^{4} - 326970 T^{5} + 50553 p T^{6} - 3118 p^{2} T^{7} + 357 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 4 T + 179 T^{2} - 96 T^{3} + 13138 T^{4} - 48712 T^{5} + 13138 p T^{6} - 96 p^{2} T^{7} + 179 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 27 T + 558 T^{2} + 8237 T^{3} + 99931 T^{4} + 998672 T^{5} + 99931 p T^{6} + 8237 p^{2} T^{7} + 558 p^{3} T^{8} + 27 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 5 T + 315 T^{2} + 612 T^{3} + 41488 T^{4} + 27302 T^{5} + 41488 p T^{6} + 612 p^{2} T^{7} + 315 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 8 T + 137 T^{2} - 852 T^{3} + 5470 T^{4} - 41656 T^{5} + 5470 p T^{6} - 852 p^{2} T^{7} + 137 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
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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

−6.08262397572684075613798284197, −5.89738352806542964796329360249, −5.89699585919009674172690492448, −5.88518483527926250900517213249, −5.68266490734444393034847797709, −4.99386838743371295420574894732, −4.98249291290856679379388056001, −4.72324148694543415708193341428, −4.65026141665517062976260240413, −4.63762632841431908586469060012, −4.08309539823270834798204444146, −3.89394826466173165547489395596, −3.70209917607951266001038280678, −3.64816454648766133126782610335, −3.48679632023936185954860294140, −3.01361664001504053558423239498, −2.75325141198962949383988410714, −2.68050635901633903133867479863, −2.52147684464407657087148595927, −2.22390540113958314242010144471, −1.66884682319572360530551967629, −1.49024632447679905921558947112, −1.38857101931261348512828429348, −1.26215011841029501593273291481, −1.07257762544020371814181271002, 1.07257762544020371814181271002, 1.26215011841029501593273291481, 1.38857101931261348512828429348, 1.49024632447679905921558947112, 1.66884682319572360530551967629, 2.22390540113958314242010144471, 2.52147684464407657087148595927, 2.68050635901633903133867479863, 2.75325141198962949383988410714, 3.01361664001504053558423239498, 3.48679632023936185954860294140, 3.64816454648766133126782610335, 3.70209917607951266001038280678, 3.89394826466173165547489395596, 4.08309539823270834798204444146, 4.63762632841431908586469060012, 4.65026141665517062976260240413, 4.72324148694543415708193341428, 4.98249291290856679379388056001, 4.99386838743371295420574894732, 5.68266490734444393034847797709, 5.88518483527926250900517213249, 5.89699585919009674172690492448, 5.89738352806542964796329360249, 6.08262397572684075613798284197

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