Properties

Degree 10
Conductor $ 2^{5} \cdot 3^{5} \cdot 11^{5} \cdot 61^{5} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s + 5·3-s + 15·4-s − 5-s − 25·6-s − 3·7-s − 35·8-s + 15·9-s + 5·10-s − 5·11-s + 75·12-s + 2·13-s + 15·14-s − 5·15-s + 70·16-s + 6·17-s − 75·18-s + 7·19-s − 15·20-s − 15·21-s + 25·22-s − 12·23-s − 175·24-s − 12·25-s − 10·26-s + 35·27-s − 45·28-s + ⋯
L(s)  = 1  − 3.53·2-s + 2.88·3-s + 15/2·4-s − 0.447·5-s − 10.2·6-s − 1.13·7-s − 12.3·8-s + 5·9-s + 1.58·10-s − 1.50·11-s + 21.6·12-s + 0.554·13-s + 4.00·14-s − 1.29·15-s + 35/2·16-s + 1.45·17-s − 17.6·18-s + 1.60·19-s − 3.35·20-s − 3.27·21-s + 5.33·22-s − 2.50·23-s − 35.7·24-s − 2.39·25-s − 1.96·26-s + 6.73·27-s − 8.50·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 11^{5} \cdot 61^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 11^{5} \cdot 61^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(10\)
\( N \)  =  \(2^{5} \cdot 3^{5} \cdot 11^{5} \cdot 61^{5}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4026} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((10,\ 2^{5} \cdot 3^{5} \cdot 11^{5} \cdot 61^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(2.046638591\)
\(L(\frac12)\)  \(\approx\)  \(2.046638591\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;61\}$,\(F_p(T)\) is a polynomial of degree 10. If $p \in \{2,\;3,\;11,\;61\}$, then $F_p(T)$ is a polynomial of degree at most 9.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + T )^{5} \)
3$C_1$ \( ( 1 - T )^{5} \)
11$C_1$ \( ( 1 + T )^{5} \)
61$C_1$ \( ( 1 + T )^{5} \)
good5$C_2 \wr S_5$ \( 1 + T + 13 T^{2} + 13 T^{3} + 4 p^{2} T^{4} + 4 p^{2} T^{5} + 4 p^{3} T^{6} + 13 p^{2} T^{7} + 13 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 3 T + 9 T^{2} + 6 T^{4} - 109 T^{5} + 6 p T^{6} + 9 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 2 T + 31 T^{2} - 48 T^{3} + 557 T^{4} - 911 T^{5} + 557 p T^{6} - 48 p^{2} T^{7} + 31 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 6 T + 54 T^{2} - 207 T^{3} + 995 T^{4} - 3429 T^{5} + 995 p T^{6} - 207 p^{2} T^{7} + 54 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 7 T + 31 T^{2} - 6 p T^{3} + 536 T^{4} - 3085 T^{5} + 536 p T^{6} - 6 p^{3} T^{7} + 31 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 12 T + 128 T^{2} + 909 T^{3} + 5835 T^{4} + 29074 T^{5} + 5835 p T^{6} + 909 p^{2} T^{7} + 128 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 4 T + 96 T^{2} - 109 T^{3} + 3365 T^{4} + 970 T^{5} + 3365 p T^{6} - 109 p^{2} T^{7} + 96 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + T + 133 T^{2} + 111 T^{3} + 7658 T^{4} + 5020 T^{5} + 7658 p T^{6} + 111 p^{2} T^{7} + 133 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 3 T + 103 T^{2} - 174 T^{3} + 5588 T^{4} - 8037 T^{5} + 5588 p T^{6} - 174 p^{2} T^{7} + 103 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 3 T + 112 T^{2} + 213 T^{3} + 6631 T^{4} + 7488 T^{5} + 6631 p T^{6} + 213 p^{2} T^{7} + 112 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 24 T + 400 T^{2} - 4559 T^{3} + 41953 T^{4} - 301526 T^{5} + 41953 p T^{6} - 4559 p^{2} T^{7} + 400 p^{3} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 18 T + 261 T^{2} - 2562 T^{3} + 23795 T^{4} - 170127 T^{5} + 23795 p T^{6} - 2562 p^{2} T^{7} + 261 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 13 T + 305 T^{2} + 2764 T^{3} + 34410 T^{4} + 219955 T^{5} + 34410 p T^{6} + 2764 p^{2} T^{7} + 305 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 16 T + 187 T^{2} + 2226 T^{3} + 21689 T^{4} + 167401 T^{5} + 21689 p T^{6} + 2226 p^{2} T^{7} + 187 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 18 T + 279 T^{2} - 2904 T^{3} + 32349 T^{4} - 267409 T^{5} + 32349 p T^{6} - 2904 p^{2} T^{7} + 279 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 31 T + 687 T^{2} + 10159 T^{3} + 122096 T^{4} + 1126272 T^{5} + 122096 p T^{6} + 10159 p^{2} T^{7} + 687 p^{3} T^{8} + 31 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 8 T + 176 T^{2} - 691 T^{3} + 12623 T^{4} - 18442 T^{5} + 12623 p T^{6} - 691 p^{2} T^{7} + 176 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 32 T + 521 T^{2} - 6042 T^{3} + 60211 T^{4} - 551709 T^{5} + 60211 p T^{6} - 6042 p^{2} T^{7} + 521 p^{3} T^{8} - 32 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 8 T + 222 T^{2} - 875 T^{3} + 16877 T^{4} - 33826 T^{5} + 16877 p T^{6} - 875 p^{2} T^{7} + 222 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - T + 249 T^{2} + 215 T^{3} + 30584 T^{4} + 42508 T^{5} + 30584 p T^{6} + 215 p^{2} T^{7} + 249 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 4 T + 361 T^{2} + 1626 T^{3} + 61187 T^{4} + 232277 T^{5} + 61187 p T^{6} + 1626 p^{2} T^{7} + 361 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.97673425380646067391230394658, −4.71473970971267244223937090462, −4.56014651097961851850364056339, −4.39745633283712581779086197603, −4.26016256632928677438457130281, −3.89404157900984236342158455809, −3.78766681361391402568595764003, −3.59479738287534794253280567729, −3.56019868620478195573989182361, −3.42072649944591451556600467429, −3.01103704302308926235174790304, −2.96144650604747442980172507402, −2.94003214207033804712280615843, −2.66986207861478079118925533436, −2.66471693224979398723133324248, −2.08294961971783177456802230485, −2.00642259728775169766992514351, −1.95961061094092407107240920579, −1.87147480329203994909869001847, −1.82143010174598262334040082536, −1.00241819620386381406635628560, −0.988883588123450762761338848117, −0.966005458342928826657615941027, −0.52437709068718499791329215528, −0.27165629202035447610509418346, 0.27165629202035447610509418346, 0.52437709068718499791329215528, 0.966005458342928826657615941027, 0.988883588123450762761338848117, 1.00241819620386381406635628560, 1.82143010174598262334040082536, 1.87147480329203994909869001847, 1.95961061094092407107240920579, 2.00642259728775169766992514351, 2.08294961971783177456802230485, 2.66471693224979398723133324248, 2.66986207861478079118925533436, 2.94003214207033804712280615843, 2.96144650604747442980172507402, 3.01103704302308926235174790304, 3.42072649944591451556600467429, 3.56019868620478195573989182361, 3.59479738287534794253280567729, 3.78766681361391402568595764003, 3.89404157900984236342158455809, 4.26016256632928677438457130281, 4.39745633283712581779086197603, 4.56014651097961851850364056339, 4.71473970971267244223937090462, 4.97673425380646067391230394658

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