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Properties

Label 10-119e5-1.1-c1e5-0-0
Degree $10$
Conductor $23863536599$
Sign $1$
Analytic cond. $0.774676$
Root an. cond. $0.974792$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 2·4-s − 4·6-s − 5·7-s + 2·8-s − 2·11-s − 4·12-s + 2·13-s − 10·14-s − 2·16-s + 5·17-s + 6·19-s + 10·21-s − 4·22-s − 10·23-s − 4·24-s − 2·25-s + 4·26-s − 4·27-s − 10·28-s − 8·29-s − 7·32-s + 4·33-s + 10·34-s + 8·37-s + 12·38-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 4-s − 1.63·6-s − 1.88·7-s + 0.707·8-s − 0.603·11-s − 1.15·12-s + 0.554·13-s − 2.67·14-s − 1/2·16-s + 1.21·17-s + 1.37·19-s + 2.18·21-s − 0.852·22-s − 2.08·23-s − 0.816·24-s − 2/5·25-s + 0.784·26-s − 0.769·27-s − 1.88·28-s − 1.48·29-s − 1.23·32-s + 0.696·33-s + 1.71·34-s + 1.31·37-s + 1.94·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{5} \cdot 17^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{5} \cdot 17^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(7^{5} \cdot 17^{5}\)
Sign: $1$
Analytic conductor: \(0.774676\)
Root analytic conductor: \(0.974792\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 7^{5} \cdot 17^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.006212843\)
\(L(\frac12)\) \(\approx\) \(1.006212843\)
\(L(\frac{3}{2})\) 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)$
bad7$C_1$ \( ( 1 + T )^{5} \)
17$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - p T + p T^{2} - p T^{3} + 3 p T^{4} - 9 T^{5} + 3 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 + 2 T + 4 T^{2} + 4 p T^{3} + 22 T^{4} + 8 p T^{5} + 22 p T^{6} + 4 p^{3} T^{7} + 4 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 2 T^{2} + 18 T^{3} + 36 T^{4} + 2 T^{5} + 36 p T^{6} + 18 p^{2} T^{7} + 2 p^{3} T^{8} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 2 T + p T^{2} + 48 T^{3} + 254 T^{4} + 380 T^{5} + 254 p T^{6} + 48 p^{2} T^{7} + p^{4} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 2 T + 25 T^{2} - 48 T^{3} + 482 T^{4} - 1116 T^{5} + 482 p T^{6} - 48 p^{2} T^{7} + 25 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 6 T + 83 T^{2} - 400 T^{3} + 2974 T^{4} - 10932 T^{5} + 2974 p T^{6} - 400 p^{2} T^{7} + 83 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 10 T + 107 T^{2} + 776 T^{3} + 5010 T^{4} + 24988 T^{5} + 5010 p T^{6} + 776 p^{2} T^{7} + 107 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 8 T + 73 T^{2} + 16 p T^{3} + 3362 T^{4} + 16048 T^{5} + 3362 p T^{6} + 16 p^{3} T^{7} + 73 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 122 T^{2} - 94 T^{3} + 6464 T^{4} - 5844 T^{5} + 6464 p T^{6} - 94 p^{2} T^{7} + 122 p^{3} T^{8} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 8 T + 81 T^{2} - 752 T^{3} + 5730 T^{4} - 29360 T^{5} + 5730 p T^{6} - 752 p^{2} T^{7} + 81 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 18 T + 284 T^{2} - 3016 T^{3} + 26390 T^{4} - 186634 T^{5} + 26390 p T^{6} - 3016 p^{2} T^{7} + 284 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 8 T + 184 T^{2} - 1160 T^{3} + 14648 T^{4} - 71228 T^{5} + 14648 p T^{6} - 1160 p^{2} T^{7} + 184 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 10 T + 187 T^{2} + 1064 T^{3} + 12618 T^{4} + 53532 T^{5} + 12618 p T^{6} + 1064 p^{2} T^{7} + 187 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 4 T + 232 T^{2} - 772 T^{3} + 23144 T^{4} - 59222 T^{5} + 23144 p T^{6} - 772 p^{2} T^{7} + 232 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 8 T + 215 T^{2} - 1248 T^{3} + 20906 T^{4} - 94640 T^{5} + 20906 p T^{6} - 1248 p^{2} T^{7} + 215 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 22 T + 448 T^{2} - 5408 T^{3} + 61002 T^{4} - 490510 T^{5} + 61002 p T^{6} - 5408 p^{2} T^{7} + 448 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 16 T + 384 T^{2} - 3984 T^{3} + 52992 T^{4} - 388340 T^{5} + 52992 p T^{6} - 3984 p^{2} T^{7} + 384 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 2 T + 119 T^{2} - 304 T^{3} + 7614 T^{4} - 49636 T^{5} + 7614 p T^{6} - 304 p^{2} T^{7} + 119 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 10 T + 188 T^{2} - 708 T^{3} + 10310 T^{4} - 7906 T^{5} + 10310 p T^{6} - 708 p^{2} T^{7} + 188 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 18 T + 435 T^{2} - 5144 T^{3} + 69218 T^{4} - 585004 T^{5} + 69218 p T^{6} - 5144 p^{2} T^{7} + 435 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 12 T + 351 T^{2} + 3032 T^{3} + 51082 T^{4} + 339960 T^{5} + 51082 p T^{6} + 3032 p^{2} T^{7} + 351 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 20 T + 345 T^{2} - 3568 T^{3} + 38318 T^{4} - 310808 T^{5} + 38318 p T^{6} - 3568 p^{2} T^{7} + 345 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 12 T + 246 T^{2} - 1890 T^{3} + 26704 T^{4} - 140626 T^{5} + 26704 p T^{6} - 1890 p^{2} T^{7} + 246 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
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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

−8.377402703273289579004695071944, −8.248872921460970218436831633342, −7.997382067082151420265125893493, −7.76819068535177226189943686173, −7.59902436732142338038413310247, −7.25051521859539390629815051782, −7.08619848431718227397248397586, −6.69578923513706923496070020114, −6.45301199625848528619851078978, −6.19959789788314820696615373651, −6.05118858654806093958592741880, −5.68883989278509570580671672501, −5.59441068093551226192704092374, −5.40656277677711083732661634593, −5.25782545075674562549846485849, −4.79444240404426559479454414831, −4.22456987191560829905691626011, −4.08090296755249678835296246878, −3.78166822621131587465507209232, −3.64621661552951217949133574307, −3.31687337077359583248630073114, −2.60572208685349216938721988861, −2.49818869742691818855469877715, −2.03058086985751467070216533709, −0.75686502916089684582990509027, 0.75686502916089684582990509027, 2.03058086985751467070216533709, 2.49818869742691818855469877715, 2.60572208685349216938721988861, 3.31687337077359583248630073114, 3.64621661552951217949133574307, 3.78166822621131587465507209232, 4.08090296755249678835296246878, 4.22456987191560829905691626011, 4.79444240404426559479454414831, 5.25782545075674562549846485849, 5.40656277677711083732661634593, 5.59441068093551226192704092374, 5.68883989278509570580671672501, 6.05118858654806093958592741880, 6.19959789788314820696615373651, 6.45301199625848528619851078978, 6.69578923513706923496070020114, 7.08619848431718227397248397586, 7.25051521859539390629815051782, 7.59902436732142338038413310247, 7.76819068535177226189943686173, 7.997382067082151420265125893493, 8.248872921460970218436831633342, 8.377402703273289579004695071944

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