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Properties

Label 10-9200e5-1.1-c1e5-0-2
Degree $10$
Conductor $6.591\times 10^{19}$
Sign $-1$
Analytic cond. $2.13956\times 10^{9}$
Root an. cond. $8.57101$
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  − 2·7-s − 9-s + 11-s − 4·13-s − 4·17-s − 7·19-s − 5·23-s + 27-s + 4·29-s − 19·31-s − 15·37-s + 25·41-s − 11·47-s − 3·49-s − 3·53-s + 59-s − 5·61-s + 2·63-s + 9·67-s − 71-s + 73-s − 2·77-s + 2·79-s + 5·81-s − 45·83-s + 6·89-s + 8·91-s + ⋯
L(s)  = 1  − 0.755·7-s − 1/3·9-s + 0.301·11-s − 1.10·13-s − 0.970·17-s − 1.60·19-s − 1.04·23-s + 0.192·27-s + 0.742·29-s − 3.41·31-s − 2.46·37-s + 3.90·41-s − 1.60·47-s − 3/7·49-s − 0.412·53-s + 0.130·59-s − 0.640·61-s + 0.251·63-s + 1.09·67-s − 0.118·71-s + 0.117·73-s − 0.227·77-s + 0.225·79-s + 5/9·81-s − 4.93·83-s + 0.635·89-s + 0.838·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{20} \cdot 5^{10} \cdot 23^{5}\)
Sign: $-1$
Analytic conductor: \(2.13956\times 10^{9}\)
Root analytic conductor: \(8.57101\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{20} \cdot 5^{10} \cdot 23^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
5 \( 1 \)
23$C_1$ \( ( 1 + T )^{5} \)
good3$C_2 \wr S_5$ \( 1 + T^{2} - T^{3} - 4 T^{4} + 10 T^{5} - 4 p T^{6} - p^{2} T^{7} + p^{3} T^{8} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 2 T + p T^{2} - T^{3} + 30 T^{4} + 46 T^{5} + 30 p T^{6} - p^{2} T^{7} + p^{4} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - T + 20 T^{2} - 16 T^{3} + 227 T^{4} - 174 T^{5} + 227 p T^{6} - 16 p^{2} T^{7} + 20 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 4 T + 19 T^{2} + 133 T^{3} + 496 T^{4} + 1606 T^{5} + 496 p T^{6} + 133 p^{2} T^{7} + 19 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 4 T + 39 T^{2} + 115 T^{3} + 406 T^{4} + 1566 T^{5} + 406 p T^{6} + 115 p^{2} T^{7} + 39 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 7 T + 54 T^{2} + 352 T^{3} + 1949 T^{4} + 7810 T^{5} + 1949 p T^{6} + 352 p^{2} T^{7} + 54 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 4 T + 104 T^{2} - 500 T^{3} + 4907 T^{4} - 22264 T^{5} + 4907 p T^{6} - 500 p^{2} T^{7} + 104 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 19 T + 227 T^{2} + 2173 T^{3} + 16253 T^{4} + 98336 T^{5} + 16253 p T^{6} + 2173 p^{2} T^{7} + 227 p^{3} T^{8} + 19 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 15 T + 249 T^{2} + 2256 T^{3} + 20618 T^{4} + 125810 T^{5} + 20618 p T^{6} + 2256 p^{2} T^{7} + 249 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 25 T + 417 T^{2} - 4753 T^{3} + 42913 T^{4} - 303514 T^{5} + 42913 p T^{6} - 4753 p^{2} T^{7} + 417 p^{3} T^{8} - 25 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2$ \( ( 1 + p T^{2} )^{5} \)
47$C_2 \wr S_5$ \( 1 + 11 T + 125 T^{2} + 952 T^{3} + 5780 T^{4} + 41402 T^{5} + 5780 p T^{6} + 952 p^{2} T^{7} + 125 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 3 T + 105 T^{2} + 196 T^{3} + 8458 T^{4} + 24194 T^{5} + 8458 p T^{6} + 196 p^{2} T^{7} + 105 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - T + 153 T^{2} + 14236 T^{4} - 6606 T^{5} + 14236 p T^{6} + 153 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 5 T + 190 T^{2} + 652 T^{3} + 18257 T^{4} + 49998 T^{5} + 18257 p T^{6} + 652 p^{2} T^{7} + 190 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 9 T + 209 T^{2} - 1980 T^{3} + 24396 T^{4} - 176326 T^{5} + 24396 p T^{6} - 1980 p^{2} T^{7} + 209 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + T + 141 T^{2} - 123 T^{3} + 12967 T^{4} - 23580 T^{5} + 12967 p T^{6} - 123 p^{2} T^{7} + 141 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - T + 207 T^{2} - 20 T^{3} + 20760 T^{4} + 9066 T^{5} + 20760 p T^{6} - 20 p^{2} T^{7} + 207 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 2 T + 267 T^{2} - 760 T^{3} + 34506 T^{4} - 94092 T^{5} + 34506 p T^{6} - 760 p^{2} T^{7} + 267 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 45 T + 1199 T^{2} + 21528 T^{3} + 290714 T^{4} + 2994854 T^{5} + 290714 p T^{6} + 21528 p^{2} T^{7} + 1199 p^{3} T^{8} + 45 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 6 T + 309 T^{2} - 1624 T^{3} + 45458 T^{4} - 202212 T^{5} + 45458 p T^{6} - 1624 p^{2} T^{7} + 309 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 25 T + 446 T^{2} + 5536 T^{3} + 65833 T^{4} + 653150 T^{5} + 65833 p T^{6} + 5536 p^{2} T^{7} + 446 p^{3} T^{8} + 25 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.80412587556876950451301334818, −4.77316857366563257395799156557, −4.59638407166147456485697841567, −4.41913566498602043451403528719, −4.29075212767809678857583409335, −3.99014770252744514160015145109, −3.88506758469789832407881324015, −3.84161892818259136235845065586, −3.80123525488652340368685471074, −3.75735337424898802170193554830, −3.49282828780944601599920988642, −2.93138060591094635251267592160, −2.90661064476657773813964666777, −2.89731987989874357817541310535, −2.88611777185810853705314722059, −2.48093315460101878038243906710, −2.35998035149417267572294165892, −2.28563678076206809989920213104, −2.14102324375424043460279677511, −1.68188144223618512726653471256, −1.67060054343597940841336044208, −1.55202244628973474673365463989, −1.28462361345793510878835974880, −1.09893989760303568293088236461, −0.845160610846910166499303539918, 0, 0, 0, 0, 0, 0.845160610846910166499303539918, 1.09893989760303568293088236461, 1.28462361345793510878835974880, 1.55202244628973474673365463989, 1.67060054343597940841336044208, 1.68188144223618512726653471256, 2.14102324375424043460279677511, 2.28563678076206809989920213104, 2.35998035149417267572294165892, 2.48093315460101878038243906710, 2.88611777185810853705314722059, 2.89731987989874357817541310535, 2.90661064476657773813964666777, 2.93138060591094635251267592160, 3.49282828780944601599920988642, 3.75735337424898802170193554830, 3.80123525488652340368685471074, 3.84161892818259136235845065586, 3.88506758469789832407881324015, 3.99014770252744514160015145109, 4.29075212767809678857583409335, 4.41913566498602043451403528719, 4.59638407166147456485697841567, 4.77316857366563257395799156557, 4.80412587556876950451301334818

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