Properties

Label 10-1150e5-1.1-c3e5-0-0
Degree $10$
Conductor $2.011\times 10^{15}$
Sign $1$
Analytic cond. $1.43820\times 10^{9}$
Root an. cond. $8.23724$
Motivic weight $3$
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  − 10·2-s + 12·3-s + 60·4-s − 120·6-s − 24·7-s − 280·8-s + 10·9-s − 54·11-s + 720·12-s + 36·13-s + 240·14-s + 1.12e3·16-s + 132·17-s − 100·18-s − 50·19-s − 288·21-s + 540·22-s − 115·23-s − 3.36e3·24-s − 360·26-s − 401·27-s − 1.44e3·28-s − 104·29-s − 342·31-s − 4.03e3·32-s − 648·33-s − 1.32e3·34-s + ⋯
L(s)  = 1  − 3.53·2-s + 2.30·3-s + 15/2·4-s − 8.16·6-s − 1.29·7-s − 12.3·8-s + 0.370·9-s − 1.48·11-s + 17.3·12-s + 0.768·13-s + 4.58·14-s + 35/2·16-s + 1.88·17-s − 1.30·18-s − 0.603·19-s − 2.99·21-s + 5.23·22-s − 1.04·23-s − 28.5·24-s − 2.71·26-s − 2.85·27-s − 9.71·28-s − 0.665·29-s − 1.98·31-s − 22.2·32-s − 3.41·33-s − 6.65·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 5^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 5^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{5} \cdot 5^{10} \cdot 23^{5}\)
Sign: $1$
Analytic conductor: \(1.43820\times 10^{9}\)
Root analytic conductor: \(8.23724\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{5} \cdot 5^{10} \cdot 23^{5} ,\ ( \ : 3/2, 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(1.418993941\)
\(L(\frac12)\) \(\approx\) \(1.418993941\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{5} \)
5 \( 1 \)
23$C_1$ \( ( 1 + p T )^{5} \)
good3$C_2 \wr S_5$ \( 1 - 4 p T + 134 T^{2} - 1087 T^{3} + 2417 p T^{4} - 41846 T^{5} + 2417 p^{4} T^{6} - 1087 p^{6} T^{7} + 134 p^{9} T^{8} - 4 p^{13} T^{9} + p^{15} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 24 T + 1366 T^{2} + 32980 T^{3} + 820837 T^{4} + 16974880 T^{5} + 820837 p^{3} T^{6} + 32980 p^{6} T^{7} + 1366 p^{9} T^{8} + 24 p^{12} T^{9} + p^{15} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 54 T + 3424 T^{2} + 144076 T^{3} + 7686031 T^{4} + 257879764 T^{5} + 7686031 p^{3} T^{6} + 144076 p^{6} T^{7} + 3424 p^{9} T^{8} + 54 p^{12} T^{9} + p^{15} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 36 T + 6937 T^{2} - 165629 T^{3} + 24376333 T^{4} - 478226225 T^{5} + 24376333 p^{3} T^{6} - 165629 p^{6} T^{7} + 6937 p^{9} T^{8} - 36 p^{12} T^{9} + p^{15} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 132 T + 16345 T^{2} - 51384 p T^{3} + 48026134 T^{4} - 1330500072 T^{5} + 48026134 p^{3} T^{6} - 51384 p^{7} T^{7} + 16345 p^{9} T^{8} - 132 p^{12} T^{9} + p^{15} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 50 T + 13692 T^{2} + 841248 T^{3} + 128264943 T^{4} + 9442335036 T^{5} + 128264943 p^{3} T^{6} + 841248 p^{6} T^{7} + 13692 p^{9} T^{8} + 50 p^{12} T^{9} + p^{15} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 104 T + 82225 T^{2} + 6765071 T^{3} + 3433081629 T^{4} + 230765010655 T^{5} + 3433081629 p^{3} T^{6} + 6765071 p^{6} T^{7} + 82225 p^{9} T^{8} + 104 p^{12} T^{9} + p^{15} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 342 T + 150136 T^{2} + 34231237 T^{3} + 8659133191 T^{4} + 1443829703350 T^{5} + 8659133191 p^{3} T^{6} + 34231237 p^{6} T^{7} + 150136 p^{9} T^{8} + 342 p^{12} T^{9} + p^{15} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 380 T + 179409 T^{2} - 46016496 T^{3} + 15508890186 T^{4} - 3185202456840 T^{5} + 15508890186 p^{3} T^{6} - 46016496 p^{6} T^{7} + 179409 p^{9} T^{8} - 380 p^{12} T^{9} + p^{15} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 2 T + 196855 T^{2} - 305569 T^{3} + 20967689875 T^{4} + 212273144979 T^{5} + 20967689875 p^{3} T^{6} - 305569 p^{6} T^{7} + 196855 p^{9} T^{8} - 2 p^{12} T^{9} + p^{15} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 114 T + 124300 T^{2} + 720144 p T^{3} + 13468954915 T^{4} + 2640064991940 T^{5} + 13468954915 p^{3} T^{6} + 720144 p^{7} T^{7} + 124300 p^{9} T^{8} + 114 p^{12} T^{9} + p^{15} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 966 T + 457060 T^{2} - 172450495 T^{3} + 75604748761 T^{4} - 28845147846970 T^{5} + 75604748761 p^{3} T^{6} - 172450495 p^{6} T^{7} + 457060 p^{9} T^{8} - 966 p^{12} T^{9} + p^{15} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 308 T + 299365 T^{2} - 90156552 T^{3} + 52972517366 T^{4} - 15161755883528 T^{5} + 52972517366 p^{3} T^{6} - 90156552 p^{6} T^{7} + 299365 p^{9} T^{8} - 308 p^{12} T^{9} + p^{15} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 277 T + 629452 T^{2} + 175599211 T^{3} + 207090518097 T^{4} + 48326825924732 T^{5} + 207090518097 p^{3} T^{6} + 175599211 p^{6} T^{7} + 629452 p^{9} T^{8} + 277 p^{12} T^{9} + p^{15} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 482 T + 550865 T^{2} - 262013960 T^{3} + 189274359010 T^{4} - 88779205168716 T^{5} + 189274359010 p^{3} T^{6} - 262013960 p^{6} T^{7} + 550865 p^{9} T^{8} - 482 p^{12} T^{9} + p^{15} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 2044 T + 2927975 T^{2} - 2832341264 T^{3} + 2214692873882 T^{4} - 1334167825423400 T^{5} + 2214692873882 p^{3} T^{6} - 2832341264 p^{6} T^{7} + 2927975 p^{9} T^{8} - 2044 p^{12} T^{9} + p^{15} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 208 T + 1173808 T^{2} - 487245051 T^{3} + 606637511003 T^{4} - 294921738753238 T^{5} + 606637511003 p^{3} T^{6} - 487245051 p^{6} T^{7} + 1173808 p^{9} T^{8} - 208 p^{12} T^{9} + p^{15} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 646 T + 1103243 T^{2} - 720427937 T^{3} + 617298566975 T^{4} - 373340531194589 T^{5} + 617298566975 p^{3} T^{6} - 720427937 p^{6} T^{7} + 1103243 p^{9} T^{8} - 646 p^{12} T^{9} + p^{15} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 3444 T + 6978154 T^{2} + 9591382196 T^{3} + 9916967251057 T^{4} + 7868469225687128 T^{5} + 9916967251057 p^{3} T^{6} + 9591382196 p^{6} T^{7} + 6978154 p^{9} T^{8} + 3444 p^{12} T^{9} + p^{15} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 1218 T + 2211148 T^{2} - 2196251680 T^{3} + 2390433921151 T^{4} - 1720657201017244 T^{5} + 2390433921151 p^{3} T^{6} - 2196251680 p^{6} T^{7} + 2211148 p^{9} T^{8} - 1218 p^{12} T^{9} + p^{15} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 140 T + 773845 T^{2} + 551355896 T^{3} + 765637273810 T^{4} + 206079088743528 T^{5} + 765637273810 p^{3} T^{6} + 551355896 p^{6} T^{7} + 773845 p^{9} T^{8} - 140 p^{12} T^{9} + p^{15} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 2816 T + 5330445 T^{2} - 6570663664 T^{3} + 7321963168946 T^{4} - 6886091635546368 T^{5} + 7321963168946 p^{3} T^{6} - 6570663664 p^{6} T^{7} + 5330445 p^{9} T^{8} - 2816 p^{12} T^{9} + p^{15} T^{10} \)
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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

−5.40505950433151341447704089418, −5.37626767159165499267258448769, −5.35156852362335237628670344395, −5.34033034945285156422549769687, −5.01933729878958059436704696450, −4.32468075239228882764657638127, −4.10967081525412864880384312603, −3.95523420759330042578297492621, −3.73394930860683520945183223871, −3.64775196567623296293702466059, −3.13977889568755128130200501785, −3.08931096678818842822487063905, −3.07784742509050760355624934596, −2.81512173472425835089793273199, −2.77634941216616372440306156574, −2.44449001743607572378315952792, −2.06106805413908766697694842196, −2.02164885320718285635429006802, −1.84023290715577101721155487223, −1.74609593327426130619321526409, −1.10481531739259597429201642943, −0.817414278911168374454259977701, −0.59436912558734041442478623375, −0.45964836575715670988544389568, −0.26488055787276114857115778769, 0.26488055787276114857115778769, 0.45964836575715670988544389568, 0.59436912558734041442478623375, 0.817414278911168374454259977701, 1.10481531739259597429201642943, 1.74609593327426130619321526409, 1.84023290715577101721155487223, 2.02164885320718285635429006802, 2.06106805413908766697694842196, 2.44449001743607572378315952792, 2.77634941216616372440306156574, 2.81512173472425835089793273199, 3.07784742509050760355624934596, 3.08931096678818842822487063905, 3.13977889568755128130200501785, 3.64775196567623296293702466059, 3.73394930860683520945183223871, 3.95523420759330042578297492621, 4.10967081525412864880384312603, 4.32468075239228882764657638127, 5.01933729878958059436704696450, 5.34033034945285156422549769687, 5.35156852362335237628670344395, 5.37626767159165499267258448769, 5.40505950433151341447704089418

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