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Properties

Label 10-8016e5-1.1-c1e5-0-1
Degree $10$
Conductor $3.310\times 10^{19}$
Sign $1$
Analytic cond. $1.07441\times 10^{9}$
Root an. cond. $8.00050$
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 − 5-s + 4·7-s + 15·9-s + 7·11-s − 8·13-s − 5·15-s + 5·17-s + 20·19-s + 20·21-s − 23-s − 15·25-s + 35·27-s − 5·29-s + 18·31-s + 35·33-s − 4·35-s − 17·37-s − 40·39-s + 6·41-s + 10·43-s − 15·45-s + 3·47-s − 16·49-s + 25·51-s + 15·53-s − 7·55-s + ⋯
L(s)  = 1  + 2.88·3-s − 0.447·5-s + 1.51·7-s + 5·9-s + 2.11·11-s − 2.21·13-s − 1.29·15-s + 1.21·17-s + 4.58·19-s + 4.36·21-s − 0.208·23-s − 3·25-s + 6.73·27-s − 0.928·29-s + 3.23·31-s + 6.09·33-s − 0.676·35-s − 2.79·37-s − 6.40·39-s + 0.937·41-s + 1.52·43-s − 2.23·45-s + 0.437·47-s − 2.28·49-s + 3.50·51-s + 2.06·53-s − 0.943·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{5} \cdot 167^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{5} \cdot 167^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{20} \cdot 3^{5} \cdot 167^{5}\)
Sign: $1$
Analytic conductor: \(1.07441\times 10^{9}\)
Root analytic conductor: \(8.00050\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{20} \cdot 3^{5} \cdot 167^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(90.12641624\)
\(L(\frac12)\) \(\approx\) \(90.12641624\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{5} \)
167$C_1$ \( ( 1 - T )^{5} \)
good5$C_2 \wr S_5$ \( 1 + T + 16 T^{2} + 18 T^{3} + 132 T^{4} + 129 T^{5} + 132 p T^{6} + 18 p^{2} T^{7} + 16 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 4 T + 32 T^{2} - 83 T^{3} + 390 T^{4} - 757 T^{5} + 390 p T^{6} - 83 p^{2} T^{7} + 32 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 7 T + 52 T^{2} - 216 T^{3} + 1010 T^{4} - 289 p T^{5} + 1010 p T^{6} - 216 p^{2} T^{7} + 52 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 8 T + 64 T^{2} + 350 T^{3} + 1603 T^{4} + 6413 T^{5} + 1603 p T^{6} + 350 p^{2} T^{7} + 64 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 5 T + 3 p T^{2} - 106 T^{3} + 53 p T^{4} - 1007 T^{5} + 53 p^{2} T^{6} - 106 p^{2} T^{7} + 3 p^{4} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 20 T + 248 T^{2} - 2074 T^{3} + 13271 T^{4} - 64971 T^{5} + 13271 p T^{6} - 2074 p^{2} T^{7} + 248 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + T + 38 T^{2} + 66 T^{3} + 1278 T^{4} + 185 T^{5} + 1278 p T^{6} + 66 p^{2} T^{7} + 38 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 5 T + 74 T^{2} + 250 T^{3} + 3266 T^{4} + 10091 T^{5} + 3266 p T^{6} + 250 p^{2} T^{7} + 74 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 18 T + 273 T^{2} - 2579 T^{3} + 21028 T^{4} - 125483 T^{5} + 21028 p T^{6} - 2579 p^{2} T^{7} + 273 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 17 T + 208 T^{2} + 1432 T^{3} + 8968 T^{4} + 45509 T^{5} + 8968 p T^{6} + 1432 p^{2} T^{7} + 208 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 6 T + 165 T^{2} - 1039 T^{3} + 11876 T^{4} - 65025 T^{5} + 11876 p T^{6} - 1039 p^{2} T^{7} + 165 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 10 T + 163 T^{2} - 1031 T^{3} + 11132 T^{4} - 55845 T^{5} + 11132 p T^{6} - 1031 p^{2} T^{7} + 163 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 3 T + 172 T^{2} - 341 T^{3} + 13415 T^{4} - 19347 T^{5} + 13415 p T^{6} - 341 p^{2} T^{7} + 172 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 15 T + 200 T^{2} - 1763 T^{3} + 13441 T^{4} - 101901 T^{5} + 13441 p T^{6} - 1763 p^{2} T^{7} + 200 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 17 T + 222 T^{2} - 1631 T^{3} + 11419 T^{4} - 68367 T^{5} + 11419 p T^{6} - 1631 p^{2} T^{7} + 222 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 2 T + 158 T^{2} + 962 T^{3} + 12461 T^{4} + 93871 T^{5} + 12461 p T^{6} + 962 p^{2} T^{7} + 158 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 24 T + 332 T^{2} - 3300 T^{3} + 29991 T^{4} - 250177 T^{5} + 29991 p T^{6} - 3300 p^{2} T^{7} + 332 p^{3} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + T + 63 T^{2} + 77 T^{3} + 4051 T^{4} + 28891 T^{5} + 4051 p T^{6} + 77 p^{2} T^{7} + 63 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 2 T + 256 T^{2} - 646 T^{3} + 32717 T^{4} - 64257 T^{5} + 32717 p T^{6} - 646 p^{2} T^{7} + 256 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 6 T + 209 T^{2} - 425 T^{3} + 21440 T^{4} - 23747 T^{5} + 21440 p T^{6} - 425 p^{2} T^{7} + 209 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 7 T + 286 T^{2} - 1971 T^{3} + 41947 T^{4} - 223037 T^{5} + 41947 p T^{6} - 1971 p^{2} T^{7} + 286 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 290 T^{2} - 322 T^{3} + 42889 T^{4} - 40245 T^{5} + 42889 p T^{6} - 322 p^{2} T^{7} + 290 p^{3} T^{8} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 13 T + 373 T^{2} + 3843 T^{3} + 59509 T^{4} + 500039 T^{5} + 59509 p T^{6} + 3843 p^{2} T^{7} + 373 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.31906048904999447857328872289, −4.30822886135058043771822951719, −4.18928535779687146421630221349, −4.14049035241838347592109487880, −4.04397650234032632749436627409, −3.70967254497796209230053582564, −3.61156756790947272197091957194, −3.51010394046450360684853664127, −3.31919119943583926732141288189, −3.28862824898149453253140464664, −2.92441337861540244754509001782, −2.82467209341550757850479620830, −2.82033593599284970505030176565, −2.53080391314980104766952597997, −2.44261544789218967309838068815, −1.97983712832801252495747625962, −1.96610010391744008960858465316, −1.81763412715701809334937515256, −1.74478632832740464312474276382, −1.53431196324860714412327544369, −1.18355906615374139055701386518, −1.07167102361148881915169354920, −0.77760175403011349166285644170, −0.72272402392329533996631281260, −0.43458741052574529605558666826, 0.43458741052574529605558666826, 0.72272402392329533996631281260, 0.77760175403011349166285644170, 1.07167102361148881915169354920, 1.18355906615374139055701386518, 1.53431196324860714412327544369, 1.74478632832740464312474276382, 1.81763412715701809334937515256, 1.96610010391744008960858465316, 1.97983712832801252495747625962, 2.44261544789218967309838068815, 2.53080391314980104766952597997, 2.82033593599284970505030176565, 2.82467209341550757850479620830, 2.92441337861540244754509001782, 3.28862824898149453253140464664, 3.31919119943583926732141288189, 3.51010394046450360684853664127, 3.61156756790947272197091957194, 3.70967254497796209230053582564, 4.04397650234032632749436627409, 4.14049035241838347592109487880, 4.18928535779687146421630221349, 4.30822886135058043771822951719, 4.31906048904999447857328872289

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