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Properties

Label 10-2752e5-1.1-c1e5-0-3
Degree $10$
Conductor $1.578\times 10^{17}$
Sign $1$
Analytic cond. $5.12421\times 10^{6}$
Root an. cond. $4.68773$
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 + 9·9-s + 7·11-s + 3·13-s − 5·15-s − 2·17-s + 19·19-s − 20·21-s + 6·23-s − 7·25-s + 5·27-s − 7·29-s − 12·31-s + 35·33-s + 4·35-s − 3·37-s + 15·39-s − 10·41-s − 5·43-s − 9·45-s + 5·47-s − 5·49-s − 10·51-s + 13·53-s − 7·55-s + ⋯
L(s)  = 1  + 2.88·3-s − 0.447·5-s − 1.51·7-s + 3·9-s + 2.11·11-s + 0.832·13-s − 1.29·15-s − 0.485·17-s + 4.35·19-s − 4.36·21-s + 1.25·23-s − 7/5·25-s + 0.962·27-s − 1.29·29-s − 2.15·31-s + 6.09·33-s + 0.676·35-s − 0.493·37-s + 2.40·39-s − 1.56·41-s − 0.762·43-s − 1.34·45-s + 0.729·47-s − 5/7·49-s − 1.40·51-s + 1.78·53-s − 0.943·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 43^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 43^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{30} \cdot 43^{5}\)
Sign: $1$
Analytic conductor: \(5.12421\times 10^{6}\)
Root analytic conductor: \(4.68773\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{30} \cdot 43^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(18.49693323\)
\(L(\frac12)\) \(\approx\) \(18.49693323\)
\(L(\frac{3}{2})\) 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 \( 1 \)
43$C_1$ \( ( 1 + T )^{5} \)
good3$C_2 \wr S_5$ \( 1 - 5 T + 16 T^{2} - 40 T^{3} + 91 T^{4} - 172 T^{5} + 91 p T^{6} - 40 p^{2} T^{7} + 16 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + T + 8 T^{2} + 8 T^{3} + 7 p T^{4} + 64 T^{5} + 7 p^{2} T^{6} + 8 p^{2} T^{7} + 8 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 4 T + 3 p T^{2} + 54 T^{3} + 212 T^{4} + 452 T^{5} + 212 p T^{6} + 54 p^{2} T^{7} + 3 p^{4} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 7 T + 50 T^{2} - 229 T^{3} + 1061 T^{4} - 320 p T^{5} + 1061 p T^{6} - 229 p^{2} T^{7} + 50 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 3 T + 40 T^{2} - 149 T^{3} + 799 T^{4} - 2856 T^{5} + 799 p T^{6} - 149 p^{2} T^{7} + 40 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 2 T + 64 T^{2} + 93 T^{3} + 1859 T^{4} + 2059 T^{5} + 1859 p T^{6} + 93 p^{2} T^{7} + 64 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - p T + 200 T^{2} - 1504 T^{3} + 9011 T^{4} - 43610 T^{5} + 9011 p T^{6} - 1504 p^{2} T^{7} + 200 p^{3} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 6 T + 110 T^{2} - 497 T^{3} + 4881 T^{4} - 16503 T^{5} + 4881 p T^{6} - 497 p^{2} T^{7} + 110 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 7 T + 54 T^{2} + 2 p T^{3} - 215 T^{4} - 7812 T^{5} - 215 p T^{6} + 2 p^{3} T^{7} + 54 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 12 T + 190 T^{2} + 1447 T^{3} + 12625 T^{4} + 66507 T^{5} + 12625 p T^{6} + 1447 p^{2} T^{7} + 190 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 3 T + 94 T^{2} + 448 T^{3} + 4757 T^{4} + 24730 T^{5} + 4757 p T^{6} + 448 p^{2} T^{7} + 94 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 10 T + 148 T^{2} + 1289 T^{3} + 10543 T^{4} + 72359 T^{5} + 10543 p T^{6} + 1289 p^{2} T^{7} + 148 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 5 T + 180 T^{2} - 712 T^{3} + 14179 T^{4} - 44882 T^{5} + 14179 p T^{6} - 712 p^{2} T^{7} + 180 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 13 T + 262 T^{2} - 2463 T^{3} + 28091 T^{4} - 187856 T^{5} + 28091 p T^{6} - 2463 p^{2} T^{7} + 262 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 32 T + 663 T^{2} - 9348 T^{3} + 103130 T^{4} - 880456 T^{5} + 103130 p T^{6} - 9348 p^{2} T^{7} + 663 p^{3} T^{8} - 32 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 4 T + 223 T^{2} - 862 T^{3} + 23340 T^{4} - 77564 T^{5} + 23340 p T^{6} - 862 p^{2} T^{7} + 223 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 17 T + 274 T^{2} - 2875 T^{3} + 32537 T^{4} - 267252 T^{5} + 32537 p T^{6} - 2875 p^{2} T^{7} + 274 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 4 T + 231 T^{2} - 336 T^{3} + 22622 T^{4} - 6680 T^{5} + 22622 p T^{6} - 336 p^{2} T^{7} + 231 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 2 T + 51 T^{2} - 78 T^{3} + 5012 T^{4} - 30600 T^{5} + 5012 p T^{6} - 78 p^{2} T^{7} + 51 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 27 T + 622 T^{2} + 9008 T^{3} + 114885 T^{4} + 1081982 T^{5} + 114885 p T^{6} + 9008 p^{2} T^{7} + 622 p^{3} T^{8} + 27 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 25 T + 552 T^{2} - 7451 T^{3} + 93909 T^{4} - 874508 T^{5} + 93909 p T^{6} - 7451 p^{2} T^{7} + 552 p^{3} T^{8} - 25 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 8 T + 199 T^{2} - 1518 T^{3} + 26804 T^{4} - 150052 T^{5} + 26804 p T^{6} - 1518 p^{2} T^{7} + 199 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 30 T + 480 T^{2} + 6119 T^{3} + 76347 T^{4} + 831617 T^{5} + 76347 p T^{6} + 6119 p^{2} T^{7} + 480 p^{3} T^{8} + 30 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.24024209059405320037184205486, −5.11196047465448136474748284470, −5.01205102010230306609488091149, −4.64520548283634934357697066388, −4.50605009825710052142922482901, −4.14281191971226111277224704946, −3.92174942023084486891846164478, −3.84556898871503423165002342090, −3.76580061478436018459497287712, −3.63048754988971601924021402565, −3.38210374943429000598805511171, −3.31920676017861410263904778179, −3.15372825235927189216562406930, −3.14680277104711927118256795841, −2.96570625975012884438643421543, −2.61363129279199532142395376950, −2.29416089576172708728299517446, −2.14525466233690692793903734004, −2.05894612233467895835513881025, −1.80531816513911318359509658413, −1.41755536919377481947203567501, −1.19801832886223696873597058096, −1.07617644546271927239708482235, −0.55329063958936430283101593559, −0.45084221106358338856571492273, 0.45084221106358338856571492273, 0.55329063958936430283101593559, 1.07617644546271927239708482235, 1.19801832886223696873597058096, 1.41755536919377481947203567501, 1.80531816513911318359509658413, 2.05894612233467895835513881025, 2.14525466233690692793903734004, 2.29416089576172708728299517446, 2.61363129279199532142395376950, 2.96570625975012884438643421543, 3.14680277104711927118256795841, 3.15372825235927189216562406930, 3.31920676017861410263904778179, 3.38210374943429000598805511171, 3.63048754988971601924021402565, 3.76580061478436018459497287712, 3.84556898871503423165002342090, 3.92174942023084486891846164478, 4.14281191971226111277224704946, 4.50605009825710052142922482901, 4.64520548283634934357697066388, 5.01205102010230306609488091149, 5.11196047465448136474748284470, 5.24024209059405320037184205486

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