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Properties

Label 10-7935e5-1.1-c1e5-0-0
Degree $10$
Conductor $3.146\times 10^{19}$
Sign $1$
Analytic cond. $1.02122\times 10^{9}$
Root an. cond. $7.95998$
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 − 4-s + 5·5-s + 3·7-s − 2·8-s + 15·9-s + 6·11-s − 5·12-s − 6·13-s + 25·15-s − 16-s + 7·17-s − 4·19-s − 5·20-s + 15·21-s − 10·24-s + 15·25-s + 35·27-s − 3·28-s + 9·29-s − 7·31-s + 2·32-s + 30·33-s + 15·35-s − 15·36-s + 37-s − 30·39-s + ⋯
L(s)  = 1  + 2.88·3-s − 1/2·4-s + 2.23·5-s + 1.13·7-s − 0.707·8-s + 5·9-s + 1.80·11-s − 1.44·12-s − 1.66·13-s + 6.45·15-s − 1/4·16-s + 1.69·17-s − 0.917·19-s − 1.11·20-s + 3.27·21-s − 2.04·24-s + 3·25-s + 6.73·27-s − 0.566·28-s + 1.67·29-s − 1.25·31-s + 0.353·32-s + 5.22·33-s + 2.53·35-s − 5/2·36-s + 0.164·37-s − 4.80·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 5^{5} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 5^{5} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{5} \cdot 5^{5} \cdot 23^{10}\)
Sign: $1$
Analytic conductor: \(1.02122\times 10^{9}\)
Root analytic conductor: \(7.95998\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 3^{5} \cdot 5^{5} \cdot 23^{10} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(117.8763809\)
\(L(\frac12)\) \(\approx\) \(117.8763809\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{5} \)
5$C_1$ \( ( 1 - T )^{5} \)
23 \( 1 \)
good2$C_2 \wr S_5$ \( 1 + T^{2} + p T^{3} + p T^{4} + p T^{5} + p^{2} T^{6} + p^{3} T^{7} + p^{3} T^{8} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 3 T + 22 T^{2} - 61 T^{3} + 257 T^{4} - 548 T^{5} + 257 p T^{6} - 61 p^{2} T^{7} + 22 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 6 T + 35 T^{2} - 146 T^{3} + 630 T^{4} - 2144 T^{5} + 630 p T^{6} - 146 p^{2} T^{7} + 35 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 6 T + 35 T^{2} + 34 T^{3} - 40 T^{4} - 1360 T^{5} - 40 p T^{6} + 34 p^{2} T^{7} + 35 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 7 T + 56 T^{2} - 259 T^{3} + 1051 T^{4} - 4580 T^{5} + 1051 p T^{6} - 259 p^{2} T^{7} + 56 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 4 T + 41 T^{2} + 78 T^{3} + 1116 T^{4} + 2548 T^{5} + 1116 p T^{6} + 78 p^{2} T^{7} + 41 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 9 T + 118 T^{2} - 895 T^{3} + 6109 T^{4} - 36880 T^{5} + 6109 p T^{6} - 895 p^{2} T^{7} + 118 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 7 T + 158 T^{2} + 821 T^{3} + 9849 T^{4} + 37512 T^{5} + 9849 p T^{6} + 821 p^{2} T^{7} + 158 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - T + 64 T^{2} + 17 T^{3} + 3759 T^{4} - 2004 T^{5} + 3759 p T^{6} + 17 p^{2} T^{7} + 64 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 7 T + 176 T^{2} + 931 T^{3} + 12883 T^{4} + 52628 T^{5} + 12883 p T^{6} + 931 p^{2} T^{7} + 176 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 20 T + 283 T^{2} - 2644 T^{3} + 21870 T^{4} - 146848 T^{5} + 21870 p T^{6} - 2644 p^{2} T^{7} + 283 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 151 T^{2} - 34 T^{3} + 11686 T^{4} - 604 T^{5} + 11686 p T^{6} - 34 p^{2} T^{7} + 151 p^{3} T^{8} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 3 T + 226 T^{2} + 533 T^{3} + 22241 T^{4} + 40352 T^{5} + 22241 p T^{6} + 533 p^{2} T^{7} + 226 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - T + 92 T^{2} + 353 T^{3} + 9571 T^{4} + 664 T^{5} + 9571 p T^{6} + 353 p^{2} T^{7} + 92 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 4 T + 267 T^{2} - 1010 T^{3} + 30336 T^{4} - 93356 T^{5} + 30336 p T^{6} - 1010 p^{2} T^{7} + 267 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 5 T + 194 T^{2} + 567 T^{3} + 15589 T^{4} + 30764 T^{5} + 15589 p T^{6} + 567 p^{2} T^{7} + 194 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - T + 86 T^{2} - 345 T^{3} + 477 T^{4} - 45196 T^{5} + 477 p T^{6} - 345 p^{2} T^{7} + 86 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 6 T + 231 T^{2} + 1546 T^{3} + 27864 T^{4} + 158928 T^{5} + 27864 p T^{6} + 1546 p^{2} T^{7} + 231 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 7 T^{2} - 1312 T^{3} + 10262 T^{4} + 33856 T^{5} + 10262 p T^{6} - 1312 p^{2} T^{7} + 7 p^{3} T^{8} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 41 T + 870 T^{2} - 11929 T^{3} + 125033 T^{4} - 1156692 T^{5} + 125033 p T^{6} - 11929 p^{2} T^{7} + 870 p^{3} T^{8} - 41 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 18 T + 261 T^{2} - 2264 T^{3} + 19282 T^{4} - 119276 T^{5} + 19282 p T^{6} - 2264 p^{2} T^{7} + 261 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 26 T + 641 T^{2} - 10156 T^{3} + 138638 T^{4} - 1481188 T^{5} + 138638 p T^{6} - 10156 p^{2} T^{7} + 641 p^{3} T^{8} - 26 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.48952848752353694308995938356, −4.30818560867052606314154382240, −4.27011852977421197228051045131, −4.25235976781918941316637211070, −4.10667553413459311070644251918, −3.54742686965944367764022883970, −3.53260552424961917606167751017, −3.52103843782228482298521336712, −3.32729799597219468580788978365, −3.14210637111958450566986263420, −3.08520773450335091924265146835, −2.81031701823489934203761914622, −2.73725351874180895965363681540, −2.35221493585917482642130793908, −2.27861602187062043682800633875, −2.14418221440057435951440742831, −1.97087674734439760674171417223, −1.88261464354146609879582094540, −1.80025296937437456371452969808, −1.62374687721648749677828423893, −1.25020550139045750356896403107, −1.00559613931281874013951819507, −0.920617177770825399283069423722, −0.54622581943672072281934947693, −0.54588186362882138559795202619, 0.54588186362882138559795202619, 0.54622581943672072281934947693, 0.920617177770825399283069423722, 1.00559613931281874013951819507, 1.25020550139045750356896403107, 1.62374687721648749677828423893, 1.80025296937437456371452969808, 1.88261464354146609879582094540, 1.97087674734439760674171417223, 2.14418221440057435951440742831, 2.27861602187062043682800633875, 2.35221493585917482642130793908, 2.73725351874180895965363681540, 2.81031701823489934203761914622, 3.08520773450335091924265146835, 3.14210637111958450566986263420, 3.32729799597219468580788978365, 3.52103843782228482298521336712, 3.53260552424961917606167751017, 3.54742686965944367764022883970, 4.10667553413459311070644251918, 4.25235976781918941316637211070, 4.27011852977421197228051045131, 4.30818560867052606314154382240, 4.48952848752353694308995938356

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